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Derivation of the solution of a quadratic equation. Three useful life hacks on how to solve quadratic equations faster than using a discriminant

We continue to study the topic solving equations". We have already got acquainted with linear equations and now we are going to get acquainted with quadratic equations.

First, we will look at what a quadratic equation is, how it is written in general form, and give related definitions. After this, we will use examples to examine in detail how incomplete quadratic equations are solved. Next, we will move on to solving complete equations, obtain the root formula, get acquainted with the discriminant of a quadratic equation, and consider solutions to typical examples. Finally, let's trace the connections between the roots and coefficients.

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What is a quadratic equation? Their types

First you need to clearly understand what a quadratic equation is. Therefore, it is logical to start a conversation about quadratic equations with the definition of a quadratic equation, as well as related definitions. After this, you can consider the main types of quadratic equations: reduced and unreduced, as well as complete and incomplete equations.

Definition and examples of quadratic equations

Definition.

Quadratic equation is an equation of the form a x 2 +b x+c=0, where x is a variable, a, b and c are some numbers, and a is non-zero.

Let's say right away that quadratic equations are often called equations of the second degree. This is due to the fact that the quadratic equation is algebraic equation second degree.

The stated definition allows us to give examples of quadratic equations. So 2 x 2 +6 x+1=0, 0.2 x 2 +2.5 x+0.03=0, etc. These are quadratic equations.

Definition.

Numbers a, b and c are called coefficients of the quadratic equation a·x 2 +b·x+c=0, and coefficient a is called the first, or the highest, or the coefficient of x 2, b is the second coefficient, or the coefficient of x, and c is the free term.

For example, let's take a quadratic equation of the form 5 x 2 −2 x −3=0, here the leading coefficient is 5, the second coefficient is equal to −2, and the free term is equal to −3. Please note that when the coefficients b and/or c are negative, as in the example just given, the short form of the quadratic equation is 5 x 2 −2 x−3=0 , rather than 5 x 2 +(−2 )·x+(−3)=0 .

It is worth noting that when the coefficients a and/or b are equal to 1 or −1, they are usually not explicitly present in the quadratic equation, which is due to the peculiarities of writing such . For example, in the quadratic equation y 2 −y+3=0 the leading coefficient is one, and the coefficient of y is equal to −1.

Reduced and unreduced quadratic equations

Depending on the value of the leading coefficient, reduced and unreduced quadratic equations are distinguished. Let us give the corresponding definitions.

Definition.

A quadratic equation in which the leading coefficient is 1 is called given quadratic equation. Otherwise the quadratic equation is untouched.

According to this definition, quadratic equations x 2 −3·x+1=0, x 2 −x−2/3=0, etc. – given, in each of them the first coefficient is equal to one. A 5 x 2 −x−1=0, etc. - unreduced quadratic equations, their leading coefficients are different from 1.

From any unreduced quadratic equation, by dividing both sides by the leading coefficient, you can go to the reduced one. This action is an equivalent transformation, that is, the reduced quadratic equation obtained in this way has the same roots as the original unreduced quadratic equation, or, like it, has no roots.

Let us look at an example of how the transition from an unreduced quadratic equation to a reduced one is performed.

Example.

From the equation 3 x 2 +12 x−7=0, go to the corresponding reduced quadratic equation.

Solution.

We just need to divide both sides of the original equation by the leading coefficient 3, it is non-zero, so we can perform this action. We have (3 x 2 +12 x−7):3=0:3, which is the same, (3 x 2):3+(12 x):3−7:3=0, and then (3:3) x 2 +(12:3) x−7:3=0, from where . This is how we obtained the reduced quadratic equation, which is equivalent to the original one.

Answer:

Complete and incomplete quadratic equations

The definition of a quadratic equation contains the condition a≠0. This condition is necessary so that the equation a x 2 + b x + c = 0 is quadratic, since when a = 0 it actually becomes a linear equation of the form b x + c = 0.

As for the coefficients b and c, they can be equal to zero, both individually and together. In these cases, the quadratic equation is called incomplete.

Definition.

The quadratic equation a x 2 +b x+c=0 is called incomplete, if at least one of the coefficients b, c is equal to zero.

In its turn

Definition.

Complete quadratic equation is an equation in which all coefficients are different from zero.

Such names were not given by chance. This will become clear from the following discussions.

If the coefficient b is zero, then the quadratic equation takes the form a·x 2 +0·x+c=0, and it is equivalent to the equation a·x 2 +c=0. If c=0, that is, the quadratic equation has the form a·x 2 +b·x+0=0, then it can be rewritten as a·x 2 +b·x=0. And with b=0 and c=0 we get the quadratic equation a·x 2 =0. The resulting equations differ from the complete quadratic equation in that their left-hand sides do not contain either a term with the variable x, or a free term, or both. Hence their name - incomplete quadratic equations.

So the equations x 2 +x+1=0 and −2 x 2 −5 x+0.2=0 are examples of complete quadratic equations, and x 2 =0, −2 x 2 =0, 5 x 2 +3=0 , −x 2 −5 x=0 are incomplete quadratic equations.

Solving incomplete quadratic equations

From the information in the previous paragraph it follows that there is three types of incomplete quadratic equations:

a·x 2 =0, the coefficients b=0 and c=0 correspond to it;
a x 2 +c=0 when b=0 ;
and a·x 2 +b·x=0 when c=0.

Let us examine in order how incomplete quadratic equations of each of these types are solved.

a x 2 =0

Let's start with solving incomplete quadratic equations in which the coefficients b and c are equal to zero, that is, with equations of the form a x 2 =0. The equation a·x 2 =0 is equivalent to the equation x 2 =0, which is obtained from the original by dividing both parts by a non-zero number a. Obviously, the root of the equation x 2 =0 is zero, since 0 2 =0. This equation has no other roots, which is explained by the fact that for any non-zero number p the inequality p 2 >0 holds, which means that for p≠0 the equality p 2 =0 is never achieved.

So, the incomplete quadratic equation a·x 2 =0 has a single root x=0.

As an example, we give the solution to the incomplete quadratic equation −4 x 2 =0. It is equivalent to the equation x 2 =0, its only root is x=0, therefore, the original equation has a single root zero.

A short solution in this case can be written as follows:
−4 x 2 =0 ,
x 2 =0,
x=0 .

a x 2 +c=0

Now let's look at how incomplete quadratic equations are solved in which the coefficient b is zero and c≠0, that is, equations of the form a x 2 +c=0. We know that moving a term from one side of the equation to the other with the opposite sign, as well as dividing both sides of the equation by a non-zero number, gives an equivalent equation. Therefore, we can carry out the following equivalent transformations of the incomplete quadratic equation a x 2 +c=0:

move c to the right side, which gives the equation a x 2 =−c,
and divide both sides by a, we get .

The resulting equation allows us to draw conclusions about its roots. Depending on the values of a and c, the value of the expression can be negative (for example, if a=1 and c=2, then ) or positive (for example, if a=−2 and c=6, then ), it is not zero , since by condition c≠0. Let's look at the cases separately.

If , then the equation has no roots. This statement follows from the fact that the square of any number is a non-negative number. It follows from this that when , then for any number p the equality cannot be true.

If , then the situation with the roots of the equation is different. In this case, if we remember about , then the root of the equation immediately becomes obvious; it is the number, since . It’s easy to guess that the number is also the root of the equation, indeed, . This equation has no other roots, which can be shown, for example, by contradiction. Let's do it.

Let us denote the roots of the equation just announced as x 1 and −x 1 . Suppose that the equation has one more root x 2, different from the indicated roots x 1 and −x 1. It is known that substituting its roots into an equation instead of x turns the equation into a correct numerical equality. For x 1 and −x 1 we have , and for x 2 we have . The properties of numerical equalities allow us to perform term-by-term subtraction of correct numerical equalities, so subtracting the corresponding parts of the equalities gives x 1 2 −x 2 2 =0. The properties of operations with numbers allow us to rewrite the resulting equality as (x 1 −x 2)·(x 1 +x 2)=0. We know that the product of two numbers is equal to zero if and only if at least one of them is equal to zero. Therefore, from the resulting equality it follows that x 1 −x 2 =0 and/or x 1 +x 2 =0, which is the same, x 2 =x 1 and/or x 2 =−x 1. So we came to a contradiction, since at the beginning we said that the root of the equation x 2 is different from x 1 and −x 1. This proves that the equation has no roots other than and .

Let us summarize the information in this paragraph. The incomplete quadratic equation a x 2 +c=0 is equivalent to the equation that

has no roots if ,
has two roots and , if .

Let's consider examples of solving incomplete quadratic equations of the form a·x 2 +c=0.

Let's start with the quadratic equation 9 x 2 +7=0. After moving the free term to the right side of the equation, it will take the form 9 x 2 =−7. Dividing both sides of the resulting equation by 9, we arrive at . Since the right side has a negative number, this equation has no roots, therefore, the original incomplete quadratic equation 9 x 2 +7 = 0 has no roots.

Let's solve another incomplete quadratic equation −x 2 +9=0. We move the nine to the right side: −x 2 =−9. Now we divide both sides by −1, we get x 2 =9. On the right side there is a positive number, from which we conclude that or . Then we write down the final answer: the incomplete quadratic equation −x 2 +9=0 has two roots x=3 or x=−3.

a x 2 +b x=0

It remains to deal with the solution of the last type of incomplete quadratic equations for c=0. Incomplete quadratic equations of the form a x 2 + b x = 0 allow you to solve factorization method. Obviously, we can, located on the left side of the equation, for which it is enough to take the common factor x out of brackets. This allows us to move from the original incomplete quadratic equation to an equivalent equation of the form x·(a·x+b)=0. And this equation is equivalent to the set of two equations x=0 and a x+b=0 , the last of which is linear and has a root x=−b/a .

So, the incomplete quadratic equation a·x 2 +b·x=0 has two roots x=0 and x=−b/a.

To consolidate the material, we will analyze the solution to a specific example.

Example.

Solve the equation.

Solution.

Taking x out of brackets gives the equation . It is equivalent to two equations x=0 and . We solve the resulting linear equation: , and after dividing the mixed number by an ordinary fraction, we find . Therefore, the roots of the original equation are x=0 and .

After getting the necessary practice, the solutions of such equations can be written briefly:

Answer:

x=0 , .

Discriminant, formula for the roots of a quadratic equation

To solve quadratic equations, there is a root formula. Let's write it down formula for the roots of a quadratic equation: , Where D=b 2 −4 a c- so-called discriminant of a quadratic equation. The entry essentially means that .

It is useful to know how the root formula was obtained, and how it is applied in finding the roots of quadratic equations. Let's figure this out.

Derivation of the formula for the roots of a quadratic equation

Let us need to solve the quadratic equation a·x 2 +b·x+c=0. Let's perform some equivalent transformations:

We can divide both sides of this equation by a non-zero number a, resulting in the following quadratic equation.
Now select a complete square on its left side: . After this, the equation will take the form .
At this stage, it is possible to transfer the last two terms to the right side with the opposite sign, we have .
And let’s also transform the expression on the right side: .

As a result, we arrive at an equation that is equivalent to the original quadratic equation a·x 2 +b·x+c=0.

We have already solved equations similar in form in the previous paragraphs, when we examined. This allows us to draw the following conclusions regarding the roots of the equation:

if , then the equation has no real solutions;
if , then the equation has the form , therefore, , from which its only root is visible;
if , then or , which is the same as or , that is, the equation has two roots.

Thus, the presence or absence of roots of the equation, and therefore the original quadratic equation, depends on the sign of the expression on the right side. In turn, the sign of this expression is determined by the sign of the numerator, since the denominator 4·a 2 is always positive, that is, by the sign of the expression b 2 −4·a·c. This expression b 2 −4 a c was called discriminant of a quadratic equation and designated by the letter D. From here, the essence of the discriminant is clear - by its value and sign, it is concluded whether the quadratic equation has real roots, and if so, what is their number - one or two.

We return to the equation , rewrite it using the notation of the discriminant: . And we draw conclusions:

if D<0 , то это уравнение не имеет действительных корней;
if D=0, then this equation has a single root;
finally, if D>0, then the equation has two roots or , which can be rewritten in the form or , and after expanding and reducing the fractions to a common denominator, we get .

So we derived the formulas for the roots of the quadratic equation, they look like , where the discriminant D is calculated by the formula D=b 2 −4 a c .

With their help, with a positive discriminant, you can calculate both real roots of a quadratic equation. When the discriminant is equal to zero, both formulas give the same root value corresponding to the only solution of the quadratic equation. And with a negative discriminant, when trying to use the formula for the roots of a quadratic equation, we are faced with extracting the square root from a negative number, which takes us beyond the scope of the school curriculum. With a negative discriminant, the quadratic equation has no real roots, but has a pair complex conjugate roots, which can be found using the same root formulas we obtained.

Algorithm for solving quadratic equations using root formulas

In practice, when solving a quadratic equation, you can immediately use the root formula, with which to calculate their values. But this is more related to finding complex roots.

However, in a school algebra course, we usually talk not about complex, but about real roots of a quadratic equation. In this case, it is advisable, before using the formulas for the roots of a quadratic equation, to first find the discriminant, make sure that it is non-negative (otherwise, we can conclude that the equation does not have real roots), and only then calculate the values of the roots.

The above reasoning allows us to write algorithm for solving a quadratic equation. To solve the quadratic equation a x 2 +b x+c=0, you need to:

using the discriminant formula D=b 2 −4·a·c, calculate its value;
conclude that a quadratic equation has no real roots if the discriminant is negative;
calculate the only root of the equation using the formula if D=0;
find two real roots of a quadratic equation using the root formula if the discriminant is positive.

Here we just note that if the discriminant is equal to zero, you can also use the formula; it will give the same value as .

You can move on to examples of using the algorithm for solving quadratic equations.

Examples of solving quadratic equations

Let's consider solutions to three quadratic equations with a positive, negative and zero discriminant. Having dealt with their solution, by analogy it will be possible to solve any other quadratic equation. Let's begin.

Example.

Find the roots of the equation x 2 +2·x−6=0.

Solution.

In this case, we have the following coefficients of the quadratic equation: a=1, b=2 and c=−6. According to the algorithm, you first need to calculate the discriminant; to do this, we substitute the indicated a, b and c into the discriminant formula, we have D=b 2 −4·a·c=2 2 −4·1·(−6)=4+24=28. Since 28>0, that is, the discriminant is greater than zero, the quadratic equation has two real roots. Let's find them by the formula of roots , we get , here we can simplify the expressions obtained by doing moving the multiplier beyond the root sign followed by reduction of the fraction:

Answer:

Let's move on to the next typical example.

Example.

Solve the quadratic equation −4 x 2 +28 x−49=0 .

Solution.

We start by finding the discriminant: D=28 2 −4·(−4)·(−49)=784−784=0. Therefore, this quadratic equation has a single root, which we find as , that is,

Answer:

x=3.5.

It remains to consider solving quadratic equations with a negative discriminant.

Example.

Solve the equation 5·y 2 +6·y+2=0.

Solution.

Here are the coefficients of the quadratic equation: a=5, b=6 and c=2. We substitute these values into the discriminant formula, we have D=b 2 −4·a·c=6 2 −4·5·2=36−40=−4. The discriminant is negative, therefore, this quadratic equation has no real roots.

If you need to indicate complex roots, then we apply the well-known formula for the roots of a quadratic equation, and perform operations with complex numbers:

Answer:

there are no real roots, complex roots are: .

Let us note once again that if the discriminant of a quadratic equation is negative, then in school they usually immediately write down an answer in which they indicate that there are no real roots, and complex roots are not found.

Root formula for even second coefficients

The formula for the roots of a quadratic equation, where D=b 2 −4·a·c allows you to obtain a formula of a more compact form, allowing you to solve quadratic equations with an even coefficient for x (or simply with a coefficient having the form 2·n, for example, or 14· ln5=2·7·ln5 ). Let's get her out.

Suppose we need to solve a quadratic equation of the form a x 2 +2 n x + c=0 . Let's find its roots using the formula known to us. To do this, we calculate the discriminant D=(2 n) 2 −4 a c=4 n 2 −4 a c=4 (n 2 −a c), and then we use the root formula:

Let us denote the expression n 2 −a c as D 1 (sometimes it is denoted D "). Then the formula for the roots of the quadratic equation under consideration with the second coefficient 2 n will take the form , where D 1 =n 2 −a·c.

It is easy to see that D=4·D 1 , or D 1 =D/4 . In other words, D 1 is the fourth part of the discriminant. It is clear that the sign of D 1 is the same as the sign of D . That is, the sign D 1 is also an indicator of the presence or absence of roots of a quadratic equation.

So, to solve a quadratic equation with the second coefficient 2 n, you need

Calculate D 1 =n 2 −a·c ;
If D 1<0 , то сделать вывод, что действительных корней нет;
If D 1 =0, then calculate the only root of the equation using the formula;
If D 1 >0, then find two real roots using the formula.

Let's consider solving the example using the root formula obtained in this paragraph.

Example.

Solve the quadratic equation 5 x 2 −6 x−32=0 .

Solution.

The second coefficient of this equation can be represented as 2·(−3) . That is, you can rewrite the original quadratic equation in the form 5 x 2 +2 (−3) x−32=0, here a=5, n=−3 and c=−32, and calculate the fourth part of the discriminant: D 1 =n 2 −a c=(−3) 2 −5 (−32)=9+160=169. Since its value is positive, the equation has two real roots. We find them using the corresponding root formula:

Note that it was possible to use the usual formula for the roots of a quadratic equation, but in this case more computational work would have to be performed.

Answer:

Simplifying the form of quadratic equations

Sometimes, before starting to calculate the roots of a quadratic equation using formulas, it doesn’t hurt to ask the question: “Is it possible to simplify the form of this equation?” Agree that in terms of calculations it will be easier to solve the quadratic equation 11 x 2 −4 x−6=0 than 1100 x 2 −400 x−600=0.

Typically, simplifying the form of a quadratic equation is achieved by multiplying or dividing both sides by a certain number. For example, in the previous paragraph it was possible to simplify the equation 1100 x 2 −400 x −600=0 by dividing both sides by 100.

A similar transformation is carried out with quadratic equations, the coefficients of which are not . In this case, both sides of the equation are usually divided by the absolute values of its coefficients. For example, let's take the quadratic equation 12 x 2 −42 x+48=0. absolute values of its coefficients: GCD(12, 42, 48)= GCD(GCD(12, 42), 48)= GCD(6, 48)=6. Dividing both sides of the original quadratic equation by 6, we arrive at the equivalent quadratic equation 2 x 2 −7 x+8=0.

And multiplying both sides of a quadratic equation is usually done to get rid of fractional coefficients. In this case, multiplication is carried out by the denominators of its coefficients. For example, if both sides of the quadratic equation are multiplied by LCM(6, 3, 1)=6, then it will take the simpler form x 2 +4·x−18=0.

In conclusion of this point, we note that they almost always get rid of the minus at the highest coefficient of a quadratic equation by changing the signs of all terms, which corresponds to multiplying (or dividing) both sides by −1. For example, usually one moves from the quadratic equation −2 x 2 −3 x+7=0 to the solution 2 x 2 +3 x−7=0 .

Relationship between roots and coefficients of a quadratic equation

The formula for the roots of a quadratic equation expresses the roots of the equation through its coefficients. Based on the root formula, you can obtain other relationships between roots and coefficients.

The most well-known and applicable formulas from Vieta’s theorem are of the form and . In particular, for the given quadratic equation, the sum of the roots is equal to the second coefficient with the opposite sign, and the product of the roots is equal to the free term. For example, by looking at the form of the quadratic equation 3 x 2 −7 x + 22 = 0, we can immediately say that the sum of its roots is equal to 7/3, and the product of the roots is equal to 22/3.

Using the already written formulas, you can obtain a number of other connections between the roots and coefficients of the quadratic equation. For example, you can express the sum of the squares of the roots of a quadratic equation through its coefficients: .

Bibliography.

Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
Mordkovich A. G. Algebra. 8th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemosyne, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.

5x (x - 4) = 0

5 x = 0 or x - 4 = 0

x = ± √ 25/4

Having learned to solve equations of the first degree, of course, you want to work with others, in particular, with equations of the second degree, which are otherwise called quadratic.

Quadratic equations are equations like ax² + bx + c = 0, where the variable is x, the numbers are a, b, c, where a is not equal to zero.

If in a quadratic equation one or the other coefficient (c or b) is equal to zero, then this equation will be classified as an incomplete quadratic equation.

How to solve an incomplete quadratic equation if students have so far only been able to solve equations of the first degree? Let's consider incomplete quadratic equations of different types and simple ways to solve them.

a) If coefficient c is equal to 0, and coefficient b is not equal to zero, then ax ² + bx + 0 = 0 is reduced to an equation of the form ax ² + bx = 0.

To solve such an equation, you need to know the formula for solving an incomplete quadratic equation, which consists in factoring the left side of it and later using the condition that the product is equal to zero.

For example, 5x² - 20x = 0. We factor the left side of the equation, while performing the usual mathematical operation: taking the common factor out of brackets

5x (x - 4) = 0

We use the condition that the products are equal to zero.

5 x = 0 or x - 4 = 0

The answer will be: the first root is 0; the second root is 4.

b) If b = 0, and the free term is not equal to zero, then the equation ax ² + 0x + c = 0 is reduced to an equation of the form ax ² + c = 0. The equations are solved in two ways: a) by factoring the polynomial of the equation on the left side ; b) using the properties of the arithmetic square root. Such an equation can be solved using one of the methods, for example:

x = ± √ 25/4

x = ± 5/2. The answer will be: the first root is 5/2; the second root is equal to - 5/2.

c) If b is equal to 0 and c is equal to 0, then ax ² + 0 + 0 = 0 is reduced to an equation of the form ax ² = 0. In such an equation x will be equal to 0.

As you can see, incomplete quadratic equations can have no more than two roots.

An incomplete quadratic equation differs from classical (complete) equations in that its factors or free term are equal to zero. The graphs of such functions are parabolas. Depending on their general appearance, they are divided into 3 groups. The principles of solution for all types of equations are the same.

There is nothing complicated in determining the type of an incomplete polynomial. It is best to consider the main differences using visual examples:

If b = 0, then the equation is ax 2 + c = 0.
If c = 0, then the expression ax 2 + bx = 0 should be solved.
If b = 0 and c = 0, then the polynomial turns into an equality like ax 2 = 0.

The latter case is more of a theoretical possibility and never occurs in knowledge testing tasks, since the only correct value of the variable x in the expression is zero. In the future, methods and examples of solving incomplete quadratic equations of 1) and 2) types will be considered.

General algorithm for searching variables and examples with solutions

Regardless of the type of equation, the solution algorithm is reduced to the following steps:

Reduce the expression to a form convenient for finding roots.
Perform calculations.
Write down the answer.

The easiest way to solve incomplete equations is to factor the left side and leave a zero on the right. Thus, the formula for an incomplete quadratic equation for finding roots is reduced to calculating the value of x for each of the factors.

You can only learn how to solve it through practice, so let’s consider a specific example of finding the roots of an incomplete equation:

As you can see, in this case b = 0. Let’s factorize the left side and get the expression:

4(x – 0.5) ⋅ (x + 0.5) = 0.

Obviously, the product is equal to zero when at least one of the factors is equal to zero. The values of the variable x1 = 0.5 and (or) x2 = -0.5 meet similar requirements.

In order to easily and quickly cope with the problem of factoring a quadratic trinomial, you should remember the following formula:

If there is no free term in the expression, the problem is greatly simplified. It will be enough just to find and bracket the common denominator. For clarity, consider an example of how to solve incomplete quadratic equations of the form ax2 + bx = 0.

Let's take the variable x out of brackets and get the following expression:

x ⋅ (x + 3) = 0.

Guided by logic, we come to the conclusion that x1 = 0, and x2 = -3.

Traditional solution method and incomplete quadratic equations

What happens if you apply the discriminant formula and try to find the roots of a polynomial with coefficients equal to zero? Let's take an example from a collection of standard tasks for the Unified State Exam in Mathematics 2017, solve it using standard formulas and the factorization method.

7x 2 – 3x = 0.

Calculate the value of the discriminant: D = (-3)2 - 4 ⋅ (-7) ⋅ 0 = 9. It turns out that the polynomial has two roots:

Now, let's solve the equation by factoring and compare the results.

X ⋅ (7x + 3) = 0,

2) 7x + 3 = 0,
7x = -3,
x = -.

As you can see, both methods give the same result, but the second way to solve the equation turned out to be much easier and faster.

Vieta's theorem

But what to do with Vieta’s favorite theorem? Can this method be used when the trinomial is incomplete? Let's try to understand the aspects of reducing incomplete equations to the classical form ax2 + bx + c = 0.

In fact, it is possible to apply Vieta's theorem in this case. It is only necessary to bring the expression to a general form, replacing the missing terms with zero.

For example, with b = 0 and a = 1, in order to eliminate the possibility of confusion, the task should be written in the form: ax2 + 0 + c = 0. Then the ratio of the sum and product of the roots and factors of the polynomial can be expressed as follows:

Theoretical calculations help to get acquainted with the essence of the issue, and always require skill development in solving specific problems. Let's turn again to the reference book of typical tasks for the exam and find a suitable example:

Let us write the expression in a form convenient for applying Vieta’s theorem:

x 2 + 0 – 16 = 0.

The next step is to create a system of conditions:

Obviously, the roots of the quadratic polynomial will be x 1 = 4 and x 2 = -4.

Now, let's practice bringing the equation to its general form. Let's take the following example: 1/4× x 2 – 1 = 0

In order to apply the Vieta theorem to the expression, you need to get rid of the fraction. Multiply the left and right sides by 4, and look at the result: x2– 4 = 0. The resulting equality is ready to be solved by the Vieta theorem, but it is much easier and faster to get the answer simply by moving c = 4 to the right side of the equation: x2 = 4.

Summing up, it should be said that the best way to solve incomplete equations is factorization, which is the simplest and fastest method. If you encounter difficulties in the process of finding roots, you can refer to the traditional method of finding roots through the discriminant.

Lesson content

What is a quadratic equation and how to solve it?

We remember that an equation is an equality containing a variable whose value needs to be found.

If the variable included in the equation is raised to the second power (squared), then such an equation is called equation of the second degree or quadratic equation.

For example, the following equations are quadratic:

We solve the first of these equations, namely x 2 − 4 = 0 .

All the identical transformations that we used when solving ordinary linear equations can also be used when solving quadratic ones.

So in Eq. x 2 − 4 = 0 we move the term −4 from the left side to the right side by changing the sign:

We got the equation x 2 = 4. Earlier we said that an equation is considered solved if in one part the variable is written to the first power and its coefficient is equal to one, and the other part is equal to some number. That is, to solve the equation, it should be reduced to the form x = a, Where a- root of the equation.

We have a variable x still in the second degree, so the solution must be continued.

To solve the equation x 2 = 4 , you need to answer the question at what value x the left side will become equal to 4. Obviously, for values of 2 and −2. To derive these values, we will use the definition of square root.

Number b called the square root of a number a, If b 2 =a and is denoted as

We have a similar situation now. After all, what is x 2 = 4? Variable x in this case it is the square root of 4, since the second power x equal to 4.

Then we can write that . Calculating the right side will allow you to find out what is equal to x. The square root has two meanings: positive and negative. Then we get x= 2 and x= −2 .

Usually written like this: put a plus-minus sign in front of the square root, then find . In our case, at the stage when the expression is written, the ± sign should be placed before

Then find the arithmetic value of the square root

Expression x= ± 2 means that x= 2 and x= −2 . That is, the roots of the equation x 2 − 4 = 0 are the numbers 2 and −2. Let us write down the solution to this equation in full:

In both cases the left-hand side is zero. This means the equation is solved correctly.

Let's solve one more equation. Let it be necessary to solve a quadratic equation ( x+ 2) 2 = 25

First, let's analyze this equation. The left side is squared and it is equal to 25. What number squared is 25? Obviously, the numbers 5 and −5

That is, our task is to find x, for which the expression x+ 2 will be equal to the numbers 5 and −5. Let's write these two equations:

Let's solve both equations. These are ordinary linear equations that are easy to solve:

So the roots of the equation ( x+ 2) 2 = 25 are the numbers 3 and −7.

In this example, as in the past, you can use the definition of square root. So, in the equations ( x+ 2) 2 = 25 expression ( x+ 2) is the square root of 25. Therefore, we can first write that .

Then the right side becomes equal to ±5 . You will get two equations: x+ 2 = 5 and x+ 2 = −5. By solving each of these equations separately, we will come to the roots 3 and −7.

Let us write down the complete solution of the equation ( x+ 2) 2 = 25

From the examples considered, it can be seen that the quadratic equation has two roots. In order not to forget about the found roots, the variable x can be signed with subscripts. Thus, the root 3 can be denoted as x 1, and the root −7 through x 2

In the previous example, it was also possible to do this. The equation x 2 − 4 = 0 had roots 2 and −2. These roots could be designated as x 1 = 2 and x 2 = −2.

It also happens that a quadratic equation has only one root or no roots at all. We will consider such equations later.

Let's check the equation ( x+ 2) 2 = 25 . Let's substitute the roots 3 and −7 into it. If, with values of 3 and −7, the left side is equal to 25, then this will mean that the equation is solved correctly:

In both cases the left-hand side is 25. So the equation is correct.

The quadratic equation can be given in different forms. Its most common form looks like this:

ax 2 + bx + c= 0 ,
Where a, b, c- some numbers, x- unknown.

This is the so-called general form of the quadratic equation. In such an equation, all terms are collected in a common place (in one part), and the other part is equal to zero. Otherwise, this type of equation is called normal form of a quadratic equation.

Let equation 3 be given x 2 + 2x= 16. It contains a variable x raised to the second power, so the equation is quadratic. Let us bring this equation to a general form.

So, we need to get an equation that will be similar to the equation ax 2 + bx+ c= 0 . To do this, in Equation 3 x 2 + 2x= 16 we move 16 from the right side to the left side, changing the sign:

3x 2 + 2x − 16 = 0

We got the equation 3x 2 + 2x− 16 = 0 . In this equation a= 3 , b= 2 , c= −16 .

In a quadratic equation of the form ax 2 + bx+ c= 0 numbers a , b And c have their own names. Yes, the number a called the first or highest coefficient; number b called the second coefficient; number c called a free member.

In our case, for the equation 3x 2 + 2x− 16 = 0 the first or highest coefficient is 3; the second coefficient is the number 2; the free member is the number −16. There is another common name for numbers a, b And c — options.

So, in the equation 3x 2 + 2x− 16 = 0 the parameters are the numbers 3, 2 and −16.

In a quadratic equation, it is desirable to arrange the terms so that they are in the same order as in the normal form of the quadratic equation.

For example, if given the equation −5 + 4x 2 + x= 0 , then it is advisable to write it in normal form, that is, in the form ax 2 + bx + c= 0.

In Eq. −5 + 4x 2 + x = 0 it can be seen that the free term is −5, it should be located at the end of the left side. Member 4 x 2 contains the most significant coefficient, it should be placed first. Member x accordingly it will be located second:

A quadratic equation can take different forms depending on the case. It all depends on what the values are equal to a , b And With .

If the odds a , b And c are not equal to zero, then the quadratic equation is called complete. For example, the quadratic equation is complete 2x 2 + 6x− 8 = 0 .

If any of the coefficients is zero (that is, absent), then the equation is significantly reduced and takes on a simpler form. This quadratic equation is called incomplete. For example, the quadratic equation 2 is incomplete x 2 + 6x= 0, it contains coefficients a And b(numbers 2 and 6), but there is no free term c.

Let's consider each of these types of equations, and for each of these types we will determine our own solution method.

Let a quadratic equation be given 2x 2 + 6x− 8 = 0 . In this equation a= 2 , b= 6 , c= −8 . If b equal to zero, then the equation takes the form:

The result is equation 2 x 2 − 8 = 0 . To solve it, move −8 to the right side, changing the sign:

2x 2 = 8

To further simplify the equation, we will use the previously studied identity transformations. In this case, you can divide both parts by 2

We have the equation that we solved at the beginning of this lesson. To solve the equation x 2 = 4, you should use the definition of square root. If x 2 = 4, then. From here x= 2 and x= −2 .

So the roots of equation 2 x 2 − 8 = 0 are the numbers 2 and −2. Let us write down the solution to this equation in full:

Let's check. Let's substitute roots 2 and −2 into the original equation and perform the corresponding calculations. If at values 2 and −2 the left side is zero, then this will mean that the equation is solved correctly:

In both cases, the left side is zero, which means the equation is solved correctly.

The equation we have now solved is incomplete quadratic equation. The name speaks for itself. If the complete quadratic equation looks like ax 2 + bx+ c= 0 , then making the coefficient b zero results in an incomplete quadratic equation ax 2 + c= 0 .

We also first had a complete quadratic equation 2x 2 + 6x− 4 = 0 . But we made a coefficient b zero, that is, instead of the number 6 they put 0. As a result, the equation turned into an incomplete quadratic equation 2 x 2 − 4 = 0 .

At the beginning of this lesson we solved the quadratic equation x 2 − 4 = 0 . It is also an equation of the form ax 2 + c= 0, that is, incomplete. In him a= 1 , b= 0 , With= −4 .

Also, a quadratic equation will be incomplete if the coefficient c equal to zero.

Consider the complete quadratic equation 2x 2 + 6x− 4 = 0 . Let's make a coefficient c zero. That is, instead of the number 4 we put 0

We got quadratic equation 2 x 2 + 6x=0 which is incomplete. To solve such an equation, the variable x put out of brackets:

The result is an equation x(2x+ 6) = 0 in which to find x, at which the left side becomes equal to zero. Note that in this equation the expressions x and 2 x+ 6) are factors. One of the properties of multiplication says that the product is equal to zero if at least one of the factors is equal to zero (either the first factor or the second).

In our case, equality will be achieved if x will be equal to zero or (2 x+ 6) will be equal to zero. So let’s write it down first:

We got two equations: x= 0 and 2 x+ 6 = 0 . There is no need to solve the first equation - it has already been solved. That is, the first root is zero.

To find the second root, solve equation 2 x+ 6 = 0 . This is a simple linear equation that can be easily solved:

We see that the second root is equal to −3.

So the roots of equation 2 x 2 + 6x= 0 are the numbers 0 and −3. Let us write down the solution to this equation in full:

Let's check. Let's substitute the roots 0 and −3 into the original equation and perform the corresponding calculations. If at values 0 and −3 the left side is zero, then this will mean that the equation is solved correctly:

The next case is when the numbers b And With are equal to zero. Consider the complete quadratic equation 2x 2 + 6x− 4 = 0 . Let's make the coefficients b And c zeros. Then the equation hi looks like:

We got equation 2 x 2 = 0 . The left side is the product and the right side is zero. The product is equal to zero if at least one of the factors is equal to zero. It's obvious that x= 0 . Indeed, 2 × 0 2 = 0. Hence, 0 = 0. For other values x equality will not be achieved.

Simply put, if in a quadratic equation of the form ax 2 + bx+ c= 0 numbers b And With are equal to zero, then the root of such an equation is equal to zero.

Note that when the phrases “ b is equal to zero" or " c equals zero ", then it is implied that the parameters b or c are completely absent from the equation.

For example, if given equation 2 x 2 − 32 = 0, then we say that b= 0 . Because if compared to the full equation ax 2 + bx+ c= 0 , then you can notice that in equation 2 x 2 − 32 = 0 leading coefficient present a, equal to 2; free term −32 is present; but there is no coefficient b .

Finally, consider the complete quadratic equation ax 2 + bx+ c= 0 . As an example, let's solve the quadratic equation x 2 − 2x+ 1 = 0 .

So, we need to find x, at which the left side becomes equal to zero. Let us use the previously studied identical transformations.

First of all, notice that the left side of the equation is . If we remember how, we get on the left side ( x− 1) 2 .

Let's discuss further. The left side is squared and equals zero. What number squared is equal to zero? Obviously only 0 . Therefore, our task is to find x, for which the expression x− 1 is equal to zero. Having solved the simplest equation x− 1 = 0, you can find out what it is equal to x

The same result can be obtained if you use the square root. In equation ( x− 1) 2 = 0 expression ( x− 1) is the square root of zero. Then we can write that . In this example, there is no need to write the ± sign in front of the root, since the root of zero has only one meaning - zero. Then it turns out x− 1 = 0 . From here x= 1 .

So the root of the equation x 2 − 2x+ 1 = 0 is a unit. This equation has no other roots. In this case, we solved a quadratic equation that has only one root. This also happens.

Simple equations are not always given. Consider for example the equation x 2 + 2x− 3 = 0 .

In this case, the left side is no longer the square of the sum or difference. Therefore, we need to look for other solutions.

Note that the left side of the equation is a quadratic trinomial. Then we can try to isolate a complete square from this trinomial and see what it gives us.

Let us select a complete square from the quadratic trinomial located on the left side of the equation:

In the resulting equation, we move −4 to the right side, changing the sign:

Now let's use the square root. In equation ( x+ 1) 2 = 4 expression ( x+ 1) is the square root of 4. Then we can write that . Evaluating the right hand side will give the expression x+ 1 = ±2. This gives us two equations: x+ 1 = 2 and x+ 1 = −2, whose roots are the numbers 1 and −3

So the roots of the equation x 2 + 2x− 3 = 0 are the numbers 1 and −3.

Let's check:

Example 3. Solve the equation x 2 − 6x+ 9 = 0 , highlighting a complete square.

So the root of the equation x 2 − 6x+ 9 = 0 is 3. Let's check:

Example 4 4x 2 + 28x− 72 = 0 , selecting a complete square:

Select a complete square from the left side:

Let's move −121 from the left side to the right side, changing the sign:

Let's use the square root:

We got two simple equations: 2 x+ 7 = 11 and 2 x+ 7 = −11. Let's solve them:

Example 5. Solve the equation 2x 2 + 3x− 27 = 0

This equation is a little more complicated. When we isolate a perfect square, we represent the first term of the quadratic trinomial as the square of some expression.

So, in the previous example, the first term in the equation was 4 x 2. It could be represented as a square of expression 2 x, that is (2x) 2 = 2 2 x 2 = 4x 2 . To make sure this is correct, you can take the square root of expression 4 x 2. This is the square root of the product - it is equal to the product of the roots:

In Eq. 2x 2 + 3x− 27 = 0 the first term is 2 x 2. It cannot be represented as a square of some expression. Because there is no number whose square is equal to 2. If there were such a number, then this number would be the square root of the number 2. But the square root of the number 2 is only taken approximately. And the approximate value is not suitable for representing the number 2 as a square.

If both sides of the original equation are multiplied or divided by the same number, you get an equation equivalent to the original one. This rule also applies to quadratic equations.

Then we can divide both sides of our equation by 2. This will allow you to get rid of the two before x 2 which will subsequently give us the opportunity to select a complete square:

Let's rewrite the left side in the form of three fractions with a denominator of 2

Let's reduce the first fraction by 2. Rewrite the remaining terms on the left side without changes. The right side will still be zero:

Let's select a complete square.

When representing a term as a double product, the appearance of a factor of 2 would cause this factor and the denominator of the fraction to cancel. To prevent this from happening, the double product was multiplied by . When selecting a complete square, you should always try to ensure that the value of the original expression does not change.

Let's fold the resulting complete square:

Let's look at similar terms:

Let's move the fraction to the right side, changing the sign:

Let's use the square root. The expression is the square root of a number

To calculate the right side, we use the extraction rule:

Then our equation will take the form:

We get two equations:

Let's solve them:

So the roots of the equation 2x 2 + 3x− 27 = 0 are the numbers 3 and .

It is more convenient to leave the root in this form without dividing the numerator by the denominator. This will make checking easier.

Let's check. Let's substitute the found roots into the original equation:

In both cases, the left-hand side is zero, so the equation 2x 2 + 3x− 27 = 0 decided correctly.

Solving the equation 2x 2 + 3x− 27 = 0 , at the very beginning we divided both its parts by 2. As a result, we obtained a quadratic equation in which the coefficient before x 2 is equal to one:

This type of quadratic equation is called given quadratic equation.

Any quadratic equation of the form ax 2 + bx+ c= 0 can be made reduced. To do this, you need to divide both parts by the coefficient, which is located in front of x². In this case, both sides of the equation ax 2 + bx+ c= 0 need to be divided by a

Example 6. Solve quadratic equation 2x 2 + x+ 2 = 0

Let's make this equation reduced:

Let's select a complete square:

We got the equation , in which the square of the expression is equal to a negative number. This cannot happen, since the square of any number or expression is always positive.

Therefore there is no such meaning x, at which the left side would become equal to . So the equation has no roots.

And since the equation is equivalent to the original equation 2x 2 + x+ 2 = 0 , then it (the original equation) has no roots.

Formulas for the roots of a quadratic equation

Selecting a complete square for each quadratic equation being solved is not very convenient.

Is it possible to create universal formulas for solving quadratic equations? It turns out it is possible. Now we will do this.

Taking the literal equation as a basis ax 2 + bx+ c= 0 , and after performing some identity transformations, we can obtain formulas for deriving the roots of the quadratic equation ax 2 + bx+ c= 0 . It will be possible to substitute coefficients into these formulas a , b , With and receive ready-made solutions.

So, let's isolate the perfect square from the left side of the equation ax 2 + bx+ c= 0. First, let's make this equation reduced. Let's divide both parts into a

Now in the resulting equation we select the complete square:

Let's move the terms to the right side, changing the sign:

Let's bring the right side to a common denominator. Fractions consisting of letters lead to a common denominator. That is, the denominator of the first fraction becomes an additional factor of the second fraction, and the denominator of the second fraction becomes an additional factor of the first fraction:

In the numerator of the right side we take it out of brackets a

Let's reduce the right side by a

Since all the transformations were identical, the resulting equation has the same roots as the original equation ax 2 + bx+ c= 0.

The equation will have roots only if the right-hand side is greater than zero or equal to zero. This is because the left side is squared, and the square of any number is positive or equal to zero (if zero is squared). And what the right side will be equal to depends on what will be substituted in place of the variables a , b And c .

Since for any a not equal to zero, the denominator of the right side of the equation will always be positive, then the sign of the fraction will depend on the sign of its numerator, that is, on the expression b 2 − 4ac .

Expression b 2 − 4ac called discriminant of a quadratic equation. Discriminant is a Latin word meaning discriminator . The discriminant of a quadratic equation is denoted by the letter D

D=b 2 − 4ac

The discriminant allows you to find out in advance whether the equation has roots or not. So, in the previous task we spent a long time solving the equation 2x 2 + x+ 2 = 0 and it turned out that it had no roots. The discriminant would allow us to know in advance that there are no roots. In Eq. 2x 2 + x+ 2 = 0 odds a, b And c are equal to 2, 1 and 2 respectively. Let's substitute them into the formula D = b 2 −4ac

D = b 2 − 4ac= 1 2 − 4 × 2 × 2 = 1 − 16 = −15.

We see that D(aka b 2 − 4ac) is a negative number. Then there is no point in solving the equation 2x 2 + x+ 2 = 0, highlighting the complete square in it, because when we get to an equation of the form , it turns out that the right side will become less than zero (due to the negative discriminant). But the square of a number cannot be negative. Therefore, this equation will not have roots.

It becomes clear why ancient people considered the expression b 2 − 4ac discriminator. This expression, like an indicator, allows you to distinguish an equation that has roots from an equation that does not have roots.

So, D equals b 2 − 4ac. Let's substitute in the equation instead of the expression b 2 − 4ac letter D

If the discriminant of the original equation is less than zero ( D< 0) , то уравнение примет вид:

In this case, they say that the original equation has no roots, since the square of any number must not be negative.

If the discriminant of the original equation is greater than zero ( D> 0), then the equation will take the form:

In this case, the equation will have two roots. To derive them, we use the square root:

We got the equation . This yields two equations: And . Let's express x in each of the equations:

The resulting two equalities are the universal formulas for solving the quadratic equation ax 2 + bx+ c= 0. They are called formulas for the roots of a quadratic equation.

Most often these formulas are denoted as x 1 and x 2. That is, to calculate the first root, a formula with index 1 is used; to derive the second root, use a formula with index 2. Let’s denote our formulas in the same way:

The order in which the formulas are applied is not important.

Let's solve, for example, a quadratic equation x 2 + 2x− 8 = 0 using formulas for the roots of a quadratic equation. The coefficients of this quadratic equation are the numbers 1, 2 and −8. That is, a= 1 , b= 2 , c= −8 .

Before using the formulas for the roots of a quadratic equation, you need to find the discriminant of this equation.

Let's find the discriminant of the quadratic equation. To do this, we use the formula D=b 2 − 4 ac. Instead of variables a, b And c we will have the coefficients of the equation x 2 + 2x− 8 = 0

D=b 2 − 4ac= 2 2 − 4 × 1 × (−8) = 4 + 32 = 36

The discriminant is greater than zero. This means the equation has two roots. Now you can use the formulas for the roots of a quadratic equation:

So the roots of the equation x 2 + 2x− 8 = 0 are the numbers 2 and −4. By checking we make sure that the roots are found correctly:

Finally, consider the case when the discriminant of the quadratic equation is equal to zero. Let's go back to the equation. If the discriminant is zero, then the right side of the equation takes the form:

And in this case, the quadratic equation will have only one root. Let's use the square root:

This is another formula for deriving the square root. Let's consider its application. Previously we solved the equation x 2 − 6x+ 9 = 0 , which has one root 3. We solved it using the method of isolating a complete square. Now let's try to solve using formulas.

Let's find the discriminant of the quadratic equation. In this equation a= 1 , b= −6 , c= 9. Then, using the discriminant formula, we have:

D=b 2 − 4ac= (−6) 2 − 4 × 1 × 9 = 36 − 36 = 0

The discriminant is zero ( D= 0) . This means that the equation has only one root, and it is calculated using the formula

So the root of the equation x 2 − 6x+ 9 = 0 is the number 3.

For a quadratic equation that has one root, the formulas are also applicable And . But using each of them will give the same result.

Let's apply these two formulas to the previous equation. In both cases we get the same answer 3

If a quadratic equation has only one root, then it is advisable to use the formula rather than the formula And . This saves time and space.

Example 3. Solve the equation 5x 2 − 6x+ 1 = 0

So the roots of the equation 5x 2 − 6x+ 1 = 0 are the numbers 1 and .

Answer: 1; .

Example 4. Solve the equation x 2 + 4x+ 4 = 0

Let's find the discriminant of the quadratic equation:

The discriminant is zero. This means the equation has only one root. It is calculated by the formula

So the root of the equation x 2 + 4x+ 4 = 0 is the number −2.

Answer: -2.

Example 5. Solve the equation 3x 2 + 2x+ 4 = 0

Let's find the discriminant of the quadratic equation:

The discriminant is less than zero. This means that this equation has no roots.

Answer: no roots.

Example 6. Solve the equation (x+ 4) 2 = 3x+ 40

Let's bring this equation to normal form. On the left side is the square of the sum of two expressions. Let's break it down:

Let's move all terms from the right side to the left side, changing their signs. There will be a zero on the right side:

The discriminant is greater than zero. This means the equation has two roots. Let's use the formulas for the roots of a quadratic equation:

So the roots of the equation (x+ 4) 2 = 3x+ 40 are the numbers 3 and −8.

Answer: 3; −8.

Example 7. Solve the equation

Let's multiply both sides of this equation by 2. This will allow us to get rid of the fraction on the left side:

In the resulting equation, we move 22 from the right side to the left side, changing the sign. The right side will remain 0

Let us present similar terms on the left side:

In the resulting equation we find the discriminant:

The discriminant is greater than zero. This means the equation has two roots. Let's use the formulas for the roots of a quadratic equation:

So the roots of the equation are the numbers 23 and −1.

Answer: 23; −1.

Example 8. Solve the equation

Let's multiply both sides by the least common multiple of the denominators of both fractions. This will get rid of fractions on both sides. The least common multiple of 2 and 3 is 6. Then we get:

In the resulting equation, open the brackets in both sides:

Now let's move all the terms from the right side to the left side, changing their signs. The right side will remain 0

Let us present similar terms on the left side:

In the resulting equation we find the discriminant:

The discriminant is greater than zero. This means the equation has two roots. Let's use the formulas for the roots of a quadratic equation:

So the roots of the equation are numbers and 2.

Examples of solving quadratic equations

Example 1. Solve the equation x 2 = 81

This is the simplest quadratic equation in which you need to determine the number whose square is 81. These are the numbers 3 and −3. Let's use the square root to derive them:

Answer: 9, −9 .

Example 2. Solve the equation x 2 − 9 = 0

This is an incomplete quadratic equation. To solve it, you need to move the term −9 to the right side, changing the sign. Then we get:

Answer: 3, −3.

Example 3. Solve the equation x 2 − 9x= 0

This is an incomplete quadratic equation. To solve it, you first need to make x outside of brackets:

The left side of the equation is the product. The product is equal to zero if at least one of the factors is equal to zero.

The left side will become zero if separately x equals zero, or if the expression x− 9 is equal to zero. You will get two equations, one of which has already been solved:

Answer: 0, 9 .

Example 4. Solve the equation x 2 + 4x− 5 = 0

This is a complete quadratic equation. It can be solved by isolating a complete square or using formulas for the roots of a quadratic equation.

Let's solve this equation using formulas. Let's find the discriminant first:

D= b 2 − 4ac= 4 2 − 4 × 1 × (−5) = 16 + 20 = 36

The discriminant is greater than zero. So the equation has two roots. Let's calculate them:

Answer: 1, −5 .

Example 5. Solve the equation

Let's multiply both sides by the numbers 5, 3 and 6. This will get rid of fractions on both sides:

In the resulting equation, we transfer all terms from the right side to the left side, changing the sign. Zero will remain on the right side:

Let's look at similar terms:

Answer: 5 , .

Example 6. Solve the equation x 2 = 6

In this example, you need to use the square root:

However, the square root of 6 cannot be taken. It is extracted only approximately. The root can be extracted with a certain accuracy. Let's extract it to the nearest hundredth:

But most often the root is left in the form of a radical:

Answer:

Example 7. Solve the equation (2x+ 3) 2 + (x− 2) 2 = 13

Let's open the brackets on the left side of the equation:

In the resulting equation, we move 13 from the right side to the left side, changing the sign. Then we present similar terms:

We have obtained an incomplete quadratic equation. Let's solve it:

Answer: 0 , −1,6 .

Example 8. Solve the equation (5 + 7x)(4 − 3x) = 0

This equation can be solved in two ways. Let's look at each of them.

First way. Open the brackets and get the normal form of the quadratic equation.

Let's expand the brackets:

Let's look at similar terms:

Let us rewrite the resulting equation so that the term with the leading coefficient is placed first, the term with the second coefficient is placed second, and the free term is placed third:

To make the leading term positive, multiply both sides of the equation by −1. Then all terms of the equation will change their signs to the opposite:

Let's solve the resulting equation using the formulas for the roots of a quadratic equation:

Second way. Find values x, for which the factors on the left side of the equation are equal to zero. This method is more convenient and much shorter.

The product is equal to zero if at least one of the factors is equal to zero. In this case, the equality in the equation (5 + 7x)(4 − 3x) = 0 will be achieved if the expression (5 + 7 x) is equal to zero, or the expression (4 − 3 x) is equal to zero. Our task is to find out at what x it happens:

Examples of problem solving

Let's imagine that there is a need to build a small room with an area of 8 m2. In this case, the length of the room should be twice its width. How to determine the length and width of such a room?

Let's make an approximate drawing of this room, which illustrates the view from above:

Let us denote the width of the room by x. And the length of the room in 2 x, because according to the conditions of the problem, the length should be twice the width. A multiplier of 2 will fulfill this requirement:

The surface of the room (its floor) is a rectangle. To calculate the area of a rectangle, you need to multiply the length of the rectangle by its width. Let's do it:

2x × x

According to the conditions of the problem, the area should be 8 m 2. So expression 2 x× x should be equal to 8

2x × x = 8

The result is an equation. If you solve it, you can find the length and width of the room.

The first thing you can do is multiply on the left side of the equation:

2x 2 = 8

As a result of this transformation, the variable x moved to the second degree. And we said that if a variable included in an equation is raised to the second power (squared), then such an equation is an equation of the second degree or a quadratic equation.

To solve our quadratic equation, we will use the previously studied identical transformations. In this case, you can divide both parts by 2

Now let's use the square root. If x 2 = 4, then. From here x= 2 and x= −2 .

Through x the width of the room was indicated. The width should not be negative, so we take only the value 2 into account. This often happens when solving problems that involve a quadratic equation. The answer yields two roots, but only one of them satisfies the conditions of the problem.

And the length was indicated by 2 x. Meaning x Now we know, let’s substitute it into expression 2 x and calculate the length:

2x= 2 × 2 = 4

This means the length is 4 m and the width is 2 m. This solution satisfies the conditions of the problem, since the area of the room is 8 m2

4 × 2 = 8 m2

Answer: the length of the room is 4 m and the width is 2 m.

Example 2. A rectangular garden plot, one side of which is 10 m larger than the other, needs to be surrounded by a fence. Determine the length of the fence if it is known that the area of the plot is 1200 m2

Solution

The length of a rectangle is usually greater than its width. Let the width of the plot x meters, and the length ( x+ 10) meters. The area of the plot is 1200 m2. Let's multiply the length of the section by its width and equate it to 1200, we get the equation:

x(x+ 10) = 1200

Let's solve this equation. First, let's open the brackets on the left side:

Let's move 1200 from the right side to the left side, changing the sign. The right side will remain 0

Let's solve the resulting equation using the formulas:

Despite the fact that the quadratic equation has two roots, we take only the value 30 into account. Because width cannot be expressed as a negative number.

So through x the width of the area was indicated. It is equal to thirty meters. And the length was indicated through the expression x+ 10. Let's substitute the found value into it x and calculate the length:

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Quadratic equations. Discriminant. Solution, examples.

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And for those who “very much…”)

Types of quadratic equations

What is a quadratic equation? What does it look like? In term quadratic equation the keyword is "square". This means that in the equation Necessarily there must be an x squared. In addition to it, the equation may (or may not!) contain just X (to the first power) and just a number (free member). And there should be no X's to a power greater than two.

In mathematical terms, a quadratic equation is an equation of the form:

Here a, b and c- some numbers. b and c- absolutely any, but A– anything other than zero. For example:

Here A =1; b = 3; c = -4

Here A =2; b = -0,5; c = 2,2

Here A =-3; b = 6; c = -18

Well, you understand...

In these quadratic equations on the left there is full set members. X squared with a coefficient A, x to the first power with coefficient b And free member s.

Such quadratic equations are called full.

And if b= 0, what do we get? We have X will be lost to the first power. This happens when multiplied by zero.) It turns out, for example:

5x 2 -25 = 0,

2x 2 -6x=0,

-x 2 +4x=0

And so on. And if both coefficients b And c are equal to zero, then it’s even simpler:

2x 2 =0,

-0.3x 2 =0

Such equations where something is missing are called incomplete quadratic equations. Which is quite logical.) Please note that x squared is present in all equations.

By the way, why A can't be equal to zero? And you substitute instead A zero.) Our X squared will disappear! The equation will become linear. And the solution is completely different...

That's all the main types of quadratic equations. Complete and incomplete.

Solving quadratic equations.

Solving complete quadratic equations.

Quadratic equations are easy to solve. According to formulas and clear, simple rules. At the first stage, it is necessary to bring the given equation to a standard form, i.e. to the view:

If the equation is already given to you in this form, you do not need to do the first stage.) The main thing is to correctly determine all the coefficients, A, b And c.

The formula for finding the roots of a quadratic equation looks like this:

The expression under the root sign is called discriminant. But more about him below. As you can see, to find X, we use only a, b and c. Those. coefficients from a quadratic equation. Just carefully substitute the values a, b and c We calculate into this formula. Let's substitute with your own signs! For example, in the equation:

A =1; b = 3; c= -4. Here we write it down:

The example is almost solved:

This is the answer.

Everything is very simple. And what, you think it’s impossible to make a mistake? Well, yes, how...

The most common mistakes are confusion with sign values a, b and c. Or rather, not with their signs (where to get confused?), but with the substitution of negative values into the formula for calculating the roots. What helps here is a detailed recording of the formula with specific numbers. If there are problems with calculations, do that!

Suppose we need to solve the following example:

Here a = -6; b = -5; c = -1

Let's say you know that you rarely get answers the first time.

Well, don't be lazy. It will take about 30 seconds to write an extra line. And the number of errors will decrease sharply. So we write in detail, with all the brackets and signs:

It seems incredibly difficult to write out so carefully. But it only seems. Give it a try. Well, or choose. What's better, fast or right? Besides, I will make you happy. After a while, there will be no need to write everything down so carefully. It will work out right on its own. Especially if you use practical techniques that are described below. This evil example with a bunch of minuses can be solved easily and without errors!

But, often, quadratic equations look slightly different. For example, like this:

Did you recognize it?) Yes! This incomplete quadratic equations.

Solving incomplete quadratic equations.

They can also be solved using a general formula. You just need to understand correctly what they are equal to here. a, b and c.

Realized? In the first example a = 1; b = -4; A c? It's not there at all! Well yes, that's right. In mathematics this means that c = 0 ! That's all. Substitute zero into the formula instead c, and we will succeed. Same with the second example. Only we don’t have zero here With, A b !

But incomplete quadratic equations can be solved much more simply. Without any formulas. Let's consider the first incomplete equation. What can you do on the left side? You can take X out of brackets! Let's take it out.

And what from this? And the fact that the product equals zero if and only if any of the factors equals zero! Don't believe me? Okay, then come up with two non-zero numbers that, when multiplied, will give zero!
Does not work? Something...
Therefore, we can confidently write: x 1 = 0, x 2 = 4.

All. These will be the roots of our equation. Both are suitable. When substituting any of them into the original equation, we get the correct identity 0 = 0. As you can see, the solution is much simpler than using the general formula. Let me note, by the way, which X will be the first and which will be the second - absolutely indifferent. It is convenient to write in order, x 1- what is smaller and x 2- that which is greater.

The second equation can also be solved simply. Move 9 to the right side. We get:

All that remains is to extract the root from 9, and that’s it. It will turn out:

Also two roots . x 1 = -3, x 2 = 3.

This is how all incomplete quadratic equations are solved. Either by placing X out of brackets, or by simply moving the number to the right and then extracting the root.
It is extremely difficult to confuse these techniques. Simply because in the first case you will have to extract the root of X, which is somehow incomprehensible, and in the second case there is nothing to take out of brackets...

Discriminant. Discriminant formula.

Magic word discriminant ! Rarely a high school student has not heard this word! The phrase “we solve through a discriminant” inspires confidence and reassurance. Because there is no need to expect tricks from the discriminant! It is simple and trouble-free to use.) I remind you of the most general formula for solving any quadratic equations:

The expression under the root sign is called a discriminant. Typically the discriminant is denoted by the letter D. Discriminant formula:

D = b 2 - 4ac

And what is so remarkable about this expression? Why did it deserve a special name? What the meaning of the discriminant? After all -b, or 2a in this formula they don’t specifically call it anything... Letters and letters.

Here's the thing. When solving a quadratic equation using this formula, it is possible only three cases.

1. The discriminant is positive. This means the root can be extracted from it. Whether the root is extracted well or poorly is another question. What is important is what is extracted in principle. Then your quadratic equation has two roots. Two different solutions.

2. The discriminant is zero. Then you will have one solution. Since adding or subtracting zero in the numerator does not change anything. Strictly speaking, this is not one root, but two identical. But, in a simplified version, it is customary to talk about one solution.

3. The discriminant is negative. The square root of a negative number cannot be taken. Well, okay. This means there are no solutions.

To be honest, when simply solving quadratic equations, the concept of a discriminant is not really needed. We substitute the values of the coefficients into the formula and count. Everything happens there by itself, two roots, one, and none. However, when solving more complex tasks, without knowledge meaning and formula of the discriminant not enough. Especially in equations with parameters. Such equations are aerobatics for the State Examination and the Unified State Examination!)

So, how to solve quadratic equations through the discriminant you remembered. Or you learned, which is also not bad.) You know how to correctly determine a, b and c. Do you know how? attentively substitute them into the root formula and attentively count the result. You understand that the key word here is attentively?

Now take note of practical techniques that dramatically reduce the number of errors. The same ones that are due to inattention... For which it later becomes painful and offensive...

First appointment . Don’t be lazy before solving a quadratic equation and bring it to standard form. What does this mean?
Let's say that after all the transformations you get the following equation:

Don't rush to write the root formula! You'll almost certainly get the odds mixed up a, b and c. Construct the example correctly. First, X squared, then without square, then the free term. Like this:

And again, don’t rush! A minus in front of an X squared can really upset you. It's easy to forget... Get rid of the minus. How? Yes, as taught in the previous topic! We need to multiply the entire equation by -1. We get:

But now you can safely write down the formula for the roots, calculate the discriminant and finish solving the example. Decide for yourself. You should now have roots 2 and -1.

Reception second. Check the roots! According to Vieta's theorem. Don't be scared, I'll explain everything! Checking last thing the equation. Those. the one we used to write down the root formula. If (as in this example) the coefficient a = 1, checking the roots is easy. It is enough to multiply them. The result should be a free member, i.e. in our case -2. Please note, not 2, but -2! Free member with your sign . If it doesn’t work out, it means they’ve already screwed up somewhere. Look for the error.

If it works, you need to add the roots. Last and final check. The coefficient should be b With opposite familiar. In our case -1+2 = +1. A coefficient b, which is before the X, is equal to -1. So, everything is correct!
It’s a pity that this is so simple only for examples where x squared is pure, with a coefficient a = 1. But at least check in such equations! There will be fewer and fewer errors.

Reception third . If your equation has fractional coefficients, get rid of the fractions! Multiply the equation by a common denominator as described in the lesson "How to solve equations? Identity transformations." When working with fractions, errors keep creeping in for some reason...

By the way, I promised to simplify the evil example with a bunch of minuses. Please! Here he is.

In order not to get confused by the minuses, we multiply the equation by -1. We get:

That's all! Solving is a pleasure!

So, let's summarize the topic.

Practical Tips:

1. Before solving, we bring the quadratic equation to standard form and build it Right.

2. If there is a negative coefficient in front of the X squared, we eliminate it by multiplying the entire equation by -1.

3. If the coefficients are fractional, we eliminate the fractions by multiplying the entire equation by the corresponding factor.

4. If x squared is pure, its coefficient is equal to one, the solution can be easily verified using Vieta’s theorem. Do it!

Now we can decide.)

Solve equations:

8x 2 - 6x + 1 = 0

x 2 + 3x + 8 = 0

x 2 - 4x + 4 = 0

(x+1) 2 + x + 1 = (x+1)(x+2)

Answers (in disarray):

x 1 = 0
x 2 = 5

x 1.2 =2

x 1 = 2
x 2 = -0.5

x - any number

x 1 = -3
x 2 = 3

no solutions

x 1 = 0.25
x 2 = 0.5

Does everything fit? Great! Quadratic equations are not your headache. The first three worked, but the rest didn’t? Then the problem is not with quadratic equations. The problem is in identical transformations of equations. Take a look at the link, it's helpful.

Doesn't quite work out? Or does it not work out at all? Then Section 555 will help you. All these examples are broken down there. Shown main errors in the solution. Of course, we also talk about the use of identical transformations in solving various equations. Helps a lot!