The meaning of the quadratic equation. Solution of quadratic equations, formula of roots, examples

Formulas for the roots of a quadratic equation. The cases of real, multiple and complex roots are considered. Factorization of a square trinomial. Geometric interpretation. Examples of determining roots and factorization.

Basic Formulas

Consider the quadratic equation:
(1) .
The roots of a quadratic equation(1) are determined by the formulas:
; .
These formulas can be combined like this:
.
When the roots of the quadratic equation are known, then the polynomial of the second degree can be represented as a product of factors (factored):
.

Further, we assume that are real numbers.
Consider discriminant of a quadratic equation:
.
If the discriminant is positive, then the quadratic equation (1) has two different real roots:
; .
Then the factorization of the square trinomial has the form:
.
If the discriminant is zero, then the quadratic equation (1) has two multiple (equal) real roots:
.
Factorization:
.
If the discriminant is negative, then the quadratic equation (1) has two complex conjugate roots:
;
.
Here is the imaginary unit, ;
and are the real and imaginary parts of the roots:
; .
Then

.

Graphic interpretation

If build function graph
,
which is a parabola, then the points of intersection of the graph with the axis will be the roots of the equation
.
When , the graph intersects the abscissa axis (axis) at two points.
When , the graph touches the x-axis at one point.
When , the graph does not cross the x-axis.

Below are examples of such graphs.

Useful Formulas Related to Quadratic Equation

(f.1) ;
(f.2) ;
(f.3) .

Derivation of the formula for the roots of a quadratic equation

We perform transformations and apply formulas (f.1) and (f.3):




,
Where
; .

So, we got the formula for the polynomial of the second degree in the form:
.
From this it can be seen that the equation

performed at
And .
That is, and are the roots of the quadratic equation
.

Examples of determining the roots of a quadratic equation

Example 1

(1.1) .

Solution

.
Comparing with our equation (1.1), we find the values ​​of the coefficients:
.
Finding the discriminant:
.
Since the discriminant is positive, the equation has two real roots:
;
;
.

From here we obtain the decomposition of the square trinomial into factors:

.

Graph of the function y = 2 x 2 + 7 x + 3 crosses the x-axis at two points.

Let's plot the function
.
The graph of this function is a parabola. It crosses the x-axis (axis) at two points:
And .
These points are the roots of the original equation (1.1).

Answer

;
;
.

Example 2

Find the roots of a quadratic equation:
(2.1) .

Solution

We write the quadratic equation in general view:
.
Comparing with the original equation (2.1), we find the values ​​of the coefficients:
.
Finding the discriminant:
.
Since the discriminant is zero, the equation has two multiple (equal) roots:
;
.

Then the factorization of the trinomial has the form:
.

Graph of the function y = x 2 - 4 x + 4 touches the x-axis at one point.

Let's plot the function
.
The graph of this function is a parabola. It touches the x-axis (axis) at one point:
.
This point is the root of the original equation (2.1). Since this root is factored twice:
,
then such a root is called a multiple. That is, they consider that there are two equal roots:
.

Answer

;
.

Example 3

Find the roots of a quadratic equation:
(3.1) .

Solution

We write the quadratic equation in general form:
(1) .
Let us rewrite the original equation (3.1):
.
Comparing with (1), we find the values ​​of the coefficients:
.
Finding the discriminant:
.
The discriminant is negative, . Therefore, there are no real roots.

You can find complex roots:
;
;
.

Then


.

The graph of the function does not cross the x-axis. There are no real roots.

Let's plot the function
.
The graph of this function is a parabola. It does not cross the abscissa (axis). Therefore, there are no real roots.

Answer

There are no real roots. Complex roots:
;
;
.

Video lesson 2: Solving quadratic equations

Lecture: Quadratic equations

The equation

The equation- this is a kind of equality, in the expressions of which there is a variable.

solve the equation- means to find such a number instead of a variable that will lead it to the correct equality.

An equation may have one solution, or several, or none at all.

To solve any equation, it should be simplified as much as possible to the form:

Linear: a*x = b;

Square: a*x 2 + b*x + c = 0.

That is, any equation before solving must be converted to a standard form.

Any equation can be solved in two ways: analytical and graphical.

On the graph, the solution to the equation is considered to be the points at which the graph intersects the x-axis.

Quadratic equations

An equation can be called quadratic if, when simplified, it takes the form:

a*x 2 + b*x + c = 0.

Wherein a, b, c are coefficients of the equation that differ from zero. A "X"- root of the equation. It is believed that a quadratic equation has two roots or may not have a solution at all. The resulting roots may be the same.

"A"- the coefficient that stands in front of the root in the square.

"b"- stands before the unknown in the first degree.

"With"- free term of the equation.

If, for example, we have an equation of the form:

2x 2 -5x+3=0

In it, "2" is the coefficient at the highest term of the equation, "-5" is the second coefficient, and "3" is the free term.

Solving a quadratic equation

There are many ways to solve a quadratic equation. However, in the school mathematics course, the solution is studied using the Vieta theorem, as well as using the discriminant.

Discriminant solution:

When solving with this method it is necessary to calculate the discriminant according to the formula:

If during the calculations you got that the discriminant is less than zero, this means that this equation has no solutions.

If the discriminant is zero, then the equation has two identical solutions. In this case, the polynomial can be collapsed according to the abbreviated multiplication formula into the square of the sum or difference. Then solve it like a linear equation. Or use the formula:

If the discriminant is greater than zero, then the following method must be used:

Vieta's theorem

If the equation is reduced, that is, the coefficient at the highest term is equal to one, then you can use Vieta's theorem.

So let's say the equation is:

The roots of the equation are found as follows:

Incomplete quadratic equation

There are several options for obtaining an incomplete quadratic equation, the form of which depends on the presence of coefficients.

1. If the second and third coefficients are equal to zero (b=0, c=0), then the quadratic equation will look like:

This equation will have only decision. Equality will only be true if the solution to the equation is zero.

In continuation of the topic “Solving Equations”, the material in this article will introduce you to quadratic equations.

Let's consider everything in detail: the essence and notation of a quadratic equation, set related terms, analyze the scheme for solving incomplete and complete equations, get acquainted with the formula of roots and the discriminant, establish connections between roots and coefficients, and of course we will give a visual solution of practical examples.

Yandex.RTB R-A-339285-1

Quadratic equation, its types

Definition 1

Quadratic equation is the equation written as a x 2 + b x + c = 0, Where x– variable, a , b and c are some numbers, while a is not zero.

Often, quadratic equations are also called equations of the second degree, since in fact a quadratic equation is an algebraic equation of the second degree.

Let's give an example to illustrate the given definition: 9 x 2 + 16 x + 2 = 0 ; 7, 5 x 2 + 3, 1 x + 0, 11 = 0, etc. are quadratic equations.

Definition 2

Numbers a , b and c are the coefficients of the quadratic equation a x 2 + b x + c = 0, while the coefficient a is called the first, or senior, or coefficient at x 2, b - the second coefficient, or coefficient at x, A c called a free member.

For example, in the quadratic equation 6 x 2 - 2 x - 11 = 0 the highest coefficient is 6 , the second coefficient is − 2 , and the free term is equal to − 11 . Let us pay attention to the fact that when the coefficients b and/or c are negative, then short form records of the form 6 x 2 - 2 x - 11 = 0, but not 6 x 2 + (− 2) x + (− 11) = 0.

Let us also clarify this aspect: if the coefficients a and/or b equal 1 or − 1 , then they may not take an explicit part in writing the quadratic equation, which is explained by the peculiarities of writing the indicated numerical coefficients. For example, in the quadratic equation y 2 − y + 7 = 0 the senior coefficient is 1 and the second coefficient is − 1 .

Reduced and non-reduced quadratic equations

According to the value of the first coefficient, quadratic equations are divided into reduced and non-reduced.

Definition 3

Reduced quadratic equation is a quadratic equation where the leading coefficient is 1 . For other values ​​of the leading coefficient, the quadratic equation is unreduced.

Here are some examples: quadratic equations x 2 − 4 · x + 3 = 0 , x 2 − x − 4 5 = 0 are reduced, in each of which the leading coefficient is 1 .

9 x 2 - x - 2 = 0- unreduced quadratic equation, where the first coefficient is different from 1 .

Any unreduced quadratic equation can be converted into a reduced equation by dividing both its parts by the first coefficient (equivalent transformation). The transformed equation will have the same roots as the given non-reduced equation or will also have no roots at all.

Consideration case study will allow us to visually demonstrate the transition from an unreduced quadratic equation to a reduced one.

Example 1

Given the equation 6 x 2 + 18 x − 7 = 0 . It is necessary to convert the original equation into the reduced form.

Solution

According to the above scheme, we divide both parts of the original equation by the leading coefficient 6 . Then we get: (6 x 2 + 18 x - 7) : 3 = 0: 3, and this is the same as: (6 x 2) : 3 + (18 x) : 3 − 7: 3 = 0 and further: (6: 6) x 2 + (18: 6) x − 7: 6 = 0 . From here: x 2 + 3 x - 1 1 6 = 0 . Thus, an equation equivalent to the given one is obtained.

Answer: x 2 + 3 x - 1 1 6 = 0 .

Complete and incomplete quadratic equations

Let us turn to the definition of a quadratic equation. In it, we specified that a ≠ 0. A similar condition is necessary for the equation a x 2 + b x + c = 0 was exactly square, since a = 0 it essentially transforms into a linear equation b x + c = 0.

In the case where the coefficients b And c are equal to zero (which is possible, both individually and jointly), the quadratic equation is called incomplete.

Definition 4

Incomplete quadratic equation is a quadratic equation a x 2 + b x + c \u003d 0, where at least one of the coefficients b And c(or both) is zero.

Complete quadratic equation is a quadratic equation in which all numerical coefficients are not equal to zero.

Let's discuss why the types of quadratic equations are given precisely such names.

For b = 0, the quadratic equation takes the form a x 2 + 0 x + c = 0, which is the same as a x 2 + c = 0. At c = 0 the quadratic equation is written as a x 2 + b x + 0 = 0, which is equivalent a x 2 + b x = 0. At b = 0 And c = 0 the equation will take the form a x 2 = 0. The equations that we have obtained differ from the full quadratic equation in that their left-hand sides do not contain either a term with the variable x, or a free term, or both at once. Actually, this fact gave the name to this type of equations - incomplete.

For example, x 2 + 3 x + 4 = 0 and − 7 x 2 − 2 x + 1, 3 = 0 are complete quadratic equations; x 2 \u003d 0, − 5 x 2 \u003d 0; 11 x 2 + 2 = 0 , − x 2 − 6 x = 0 are incomplete quadratic equations.

Solving incomplete quadratic equations

The definition given above makes it possible to distinguish the following types of incomplete quadratic equations:

• a x 2 = 0, coefficients correspond to such an equation b = 0 and c = 0 ;
• a x 2 + c \u003d 0 for b \u003d 0;
• a x 2 + b x = 0 for c = 0 .

Consider successively the solution of each type of incomplete quadratic equation.

Solution of the equation a x 2 \u003d 0

As already mentioned above, such an equation corresponds to the coefficients b And c, equal to zero. The equation a x 2 = 0 can be converted into an equivalent equation x2 = 0, which we get by dividing both sides of the original equation by the number a, not equal to zero. The obvious fact is that the root of the equation x2 = 0 is zero because 0 2 = 0 . This equation has no other roots, which is explained by the properties of the degree: for any number p , not equal to zero, the inequality is true p2 > 0, from which it follows that when p ≠ 0 equality p2 = 0 will never be reached.

Definition 5

Thus, for the incomplete quadratic equation a x 2 = 0, there is a unique root x=0.

Example 2

For example, let's solve an incomplete quadratic equation − 3 x 2 = 0. It is equivalent to the equation x2 = 0, its only root is x=0, then the original equation has a single root - zero.

The solution is summarized as follows:

− 3 x 2 \u003d 0, x 2 \u003d 0, x \u003d 0.

Solution of the equation a x 2 + c \u003d 0

Next in line is the solution of incomplete quadratic equations, where b \u003d 0, c ≠ 0, that is, equations of the form a x 2 + c = 0. Let's transform this equation by transferring the term from one side of the equation to the other, changing the sign to the opposite and dividing both sides of the equation by a number that is not equal to zero:

• endure c to the right side, which gives the equation a x 2 = − c;
• divide both sides of the equation by a, we get as a result x = - c a .

Our transformations are equivalent, respectively, the resulting equation is also equivalent to the original one, and this fact makes it possible to draw a conclusion about the roots of the equation. From what are the values a And c depends on the value of the expression - c a: it can have a minus sign (for example, if a = 1 And c = 2, then - c a = - 2 1 = - 2) or a plus sign (for example, if a = -2 And c=6, then - c a = - 6 - 2 = 3); it is not equal to zero because c ≠ 0. Let us dwell in more detail on situations when - c a< 0 и - c a > 0 .

In the case when - c a< 0 , уравнение x 2 = - c a не будет иметь корней. Утверждая это, мы опираемся на то, что квадратом любого числа является число неотрицательное. Из сказанного следует, что при - c a < 0 ни для какого числа p equality p 2 = - c a cannot be true.

Everything is different when - c a > 0: remember the square root, and it will become obvious that the root of the equation x 2 \u003d - c a will be the number - c a, since - c a 2 \u003d - c a. It is easy to understand that the number - - c a - is also the root of the equation x 2 = - c a: indeed, - - c a 2 = - c a .

The equation will have no other roots. We can demonstrate this using the opposite method. First, let's set the notation of the roots found above as x 1 And − x 1. Let's assume that the equation x 2 = - c a also has a root x2, which is different from the roots x 1 And − x 1. We know that by substituting into the equation instead of x its roots, we transform the equation into a fair numerical equality.

For x 1 And − x 1 write: x 1 2 = - c a , and for x2- x 2 2 \u003d - c a. Based on the properties of numerical equalities, we subtract one true equality from another term by term, which will give us: x 1 2 − x 2 2 = 0. Use the properties of number operations to rewrite the last equality as (x 1 - x 2) (x 1 + x 2) = 0. It is known that the product of two numbers is zero if and only if at least one of the numbers is zero. From what has been said, it follows that x1 − x2 = 0 and/or x1 + x2 = 0, which is the same x2 = x1 and/or x 2 = − x 1. An obvious contradiction arose, because at first it was agreed that the root of the equation x2 differs from x 1 And − x 1. So, we have proved that the equation has no other roots than x = - c a and x = - - c a .

We summarize all the arguments above.

Definition 6

Incomplete quadratic equation a x 2 + c = 0 is equivalent to the equation x 2 = - c a , which:

• will not have roots at - c a< 0 ;
• will have two roots x = - c a and x = - - c a when - c a > 0 .

Let us give examples of solving equations a x 2 + c = 0.

Example 3

Given a quadratic equation 9 x 2 + 7 = 0 . It is necessary to find its solution.

Solution

We transfer the free term to the right side of the equation, then the equation will take the form 9 x 2 \u003d - 7.
We divide both sides of the resulting equation by 9 , we come to x 2 = - 7 9 . On the right side we see a number with a minus sign, which means: the given equation has no roots. Then the original incomplete quadratic equation 9 x 2 + 7 = 0 will not have roots.

Answer: the equation 9 x 2 + 7 = 0 has no roots.

Example 4

It is necessary to solve the equation − x2 + 36 = 0.

Solution

Let's move 36 to the right side: − x 2 = − 36.
Let's divide both parts into − 1 , we get x2 = 36. On the right side - positive number, hence it can be concluded that x = 36 or x = - 36 .
We extract the root and write the final result: an incomplete quadratic equation − x2 + 36 = 0 has two roots x=6 or x = -6.

Answer: x=6 or x = -6.

Solution of the equation a x 2 +b x=0

Let us analyze the third kind of incomplete quadratic equations, when c = 0. To find a solution to an incomplete quadratic equation a x 2 + b x = 0, we use the factorization method. Let us factorize the polynomial, which is on the left side of the equation, taking the common factor out of brackets x. This step will make it possible to transform the original incomplete quadratic equation into its equivalent x (a x + b) = 0. And this equation, in turn, is equivalent to the set of equations x=0 And a x + b = 0. The equation a x + b = 0 linear, and its root: x = − b a.

Definition 7

Thus, the incomplete quadratic equation a x 2 + b x = 0 will have two roots x=0 And x = − b a.

Let's consolidate the material with an example.

Example 5

It is necessary to find the solution of the equation 2 3 · x 2 - 2 2 7 · x = 0 .

Solution

Let's take out x outside the brackets and get the equation x · 2 3 · x - 2 2 7 = 0 . This equation is equivalent to the equations x=0 and 2 3 x - 2 2 7 = 0 . Now you should solve the resulting linear equation: 2 3 · x = 2 2 7 , x = 2 2 7 2 3 .

Briefly, we write the solution of the equation as follows:

2 3 x 2 - 2 2 7 x = 0 x 2 3 x - 2 2 7 = 0

x = 0 or 2 3 x - 2 2 7 = 0

x = 0 or x = 3 3 7

Answer: x = 0 , x = 3 3 7 .

Discriminant, formula of the roots of a quadratic equation

To find a solution to quadratic equations, there is a root formula:

Definition 8

x = - b ± D 2 a, where D = b 2 − 4 a c is the so-called discriminant of a quadratic equation.

Writing x \u003d - b ± D 2 a essentially means that x 1 \u003d - b + D 2 a, x 2 \u003d - b - D 2 a.

It will be useful to understand how the indicated formula was derived and how to apply it.

Derivation of the formula of the roots of a quadratic equation

Suppose we are faced with the task of solving a quadratic equation a x 2 + b x + c = 0. Let's carry out a number of equivalent transformations:

• divide both sides of the equation by the number a, different from zero, we obtain the reduced quadratic equation: x 2 + b a x + c a \u003d 0;
• single out full square on the left side of the resulting equation:
  x 2 + b a x + c a = x 2 + 2 b 2 a x + b 2 a 2 - b 2 a 2 + c a = = x + b 2 a 2 - b 2 a 2 + c a
  After this, the equation will take the form: x + b 2 a 2 - b 2 a 2 + c a \u003d 0;
• now it is possible to transfer the last two terms to the right side, changing the sign to the opposite, after which we get: x + b 2 · a 2 = b 2 · a 2 - c a ;
• finally, we transform the expression written on the right side of the last equality:
  b 2 a 2 - c a \u003d b 2 4 a 2 - c a \u003d b 2 4 a 2 - 4 a c 4 a 2 \u003d b 2 - 4 a c 4 a 2.

Thus, we have come to the equation x + b 2 a 2 = b 2 - 4 a c 4 a 2 , which is equivalent to the original equation a x 2 + b x + c = 0.

We discussed the solution of such equations in the previous paragraphs (the solution of incomplete quadratic equations). The experience already gained makes it possible to draw a conclusion regarding the roots of the equation x + b 2 a 2 = b 2 - 4 a c 4 a 2:

• for b 2 - 4 a c 4 a 2< 0 уравнение не имеет действительных решений;
• for b 2 - 4 · a · c 4 · a 2 = 0, the equation has the form x + b 2 · a 2 = 0, then x + b 2 · a = 0.

From here, the only root x = - b 2 · a is obvious;

• for b 2 - 4 a c 4 a 2 > 0, the correct one is: x + b 2 a = b 2 - 4 a c 4 a 2 or x = b 2 a - b 2 - 4 a c 4 a 2 , which is the same as x + - b 2 a = b 2 - 4 a c 4 a 2 or x = - b 2 a - b 2 - 4 a c 4 a 2 , i.e. the equation has two roots.

It is possible to conclude that the presence or absence of the roots of the equation x + b 2 a 2 = b 2 - 4 a c 4 a 2 (and hence the original equation) depends on the sign of the expression b 2 - 4 a c 4 · a 2 written on the right side. And the sign of this expression is given by the sign of the numerator, (the denominator 4 a 2 will always be positive), that is, the sign of the expression b 2 − 4 a c. This expression b 2 − 4 a c a name is given - the discriminant of a quadratic equation and the letter D is defined as its designation. Here you can write down the essence of the discriminant - by its value and sign, they conclude whether the quadratic equation will have real roots, and, if so, how many roots - one or two.

Let's return to the equation x + b 2 a 2 = b 2 - 4 a c 4 a 2 . Let's rewrite it using the discriminant notation: x + b 2 · a 2 = D 4 · a 2 .

Let's recap the conclusions:

Definition 9

• at D< 0 the equation has no real roots;
• at D=0 the equation has a single root x = - b 2 · a ;
• at D > 0 the equation has two roots: x \u003d - b 2 a + D 4 a 2 or x \u003d - b 2 a - D 4 a 2. Based on the properties of radicals, these roots can be written as: x \u003d - b 2 a + D 2 a or - b 2 a - D 2 a. And when we open the modules and reduce the fractions to a common denominator, we get: x \u003d - b + D 2 a, x \u003d - b - D 2 a.

So, the result of our reasoning was the derivation of the formula for the roots of the quadratic equation:

x = - b + D 2 a , x = - b - D 2 a , discriminant D calculated by the formula D = b 2 − 4 a c.

These formulas make it possible, when the discriminant is greater than zero, to determine both real roots. When the discriminant is zero, applying both formulas will give the same root as the only solution to the quadratic equation. In the case when the discriminant is negative, trying to use the quadratic root formula, we will be faced with the need to extract Square root from a negative number, which will take us beyond real numbers. With a negative discriminant, the quadratic equation will not have real roots, but a pair of complex conjugate roots is possible, determined by the same root formulas we obtained.

Algorithm for solving quadratic equations using root formulas

It is possible to solve a quadratic equation by immediately using the root formula, but basically this is done when it is necessary to find complex roots.

In the bulk of cases, the search is usually meant not for complex, but for real roots of a quadratic equation. Then it is optimal, before using the formulas for the roots of the quadratic equation, first to determine the discriminant and make sure that it is not negative (otherwise we will conclude that the equation has no real roots), and then proceed to calculate the value of the roots.

The reasoning above makes it possible to formulate an algorithm for solving a quadratic equation.

Definition 10

To solve a quadratic equation a x 2 + b x + c = 0, necessary:

• according to the formula D = b 2 − 4 a c find the value of the discriminant;
• at D< 0 сделать вывод об отсутствии у квадратного уравнения действительных корней;
• for D = 0 find the only root of the equation by the formula x = - b 2 · a ;
• for D > 0, determine two real roots of the quadratic equation by the formula x = - b ± D 2 · a.

Note that when the discriminant is zero, you can use the formula x = - b ± D 2 · a , it will give the same result as the formula x = - b 2 · a .

Consider examples.

Examples of solving quadratic equations

Let us give an example solution for different values discriminant.

Example 6

It is necessary to find the roots of the equation x 2 + 2 x - 6 = 0.

Solution

We write the numerical coefficients of the quadratic equation: a \u003d 1, b \u003d 2 and c = − 6. Next, we act according to the algorithm, i.e. Let's start calculating the discriminant, for which we substitute the coefficients a , b And c into the discriminant formula: D = b 2 − 4 a c = 2 2 − 4 1 (− 6) = 4 + 24 = 28 .

So, we got D > 0, which means that the original equation will have two real roots.
To find them, we use the root formula x \u003d - b ± D 2 · a and, substituting the appropriate values, we get: x \u003d - 2 ± 28 2 · 1. We simplify the resulting expression by taking the factor out of the sign of the root, followed by reduction of the fraction:

x = - 2 ± 2 7 2

x = - 2 + 2 7 2 or x = - 2 - 2 7 2

x = - 1 + 7 or x = - 1 - 7

Answer: x = - 1 + 7 , x = - 1 - 7 .

Example 7

It is necessary to solve a quadratic equation − 4 x 2 + 28 x − 49 = 0.

Solution

Let's define the discriminant: D = 28 2 − 4 (− 4) (− 49) = 784 − 784 = 0. With this value of the discriminant, the original equation will have only one root, determined by the formula x = - b 2 · a.

x = - 28 2 (- 4) x = 3, 5

Answer: x = 3, 5.

Example 8

It is necessary to solve the equation 5 y 2 + 6 y + 2 = 0

Solution

The numerical coefficients of this equation will be: a = 5 , b = 6 and c = 2 . We use these values ​​to find the discriminant: D = b 2 − 4 · a · c = 6 2 − 4 · 5 · 2 = 36 − 40 = − 4 . The computed discriminant is negative, so the original quadratic equation has no real roots.

In the case when the task is to indicate complex roots, we apply the root formula by performing operations with complex numbers:

x \u003d - 6 ± - 4 2 5,

x \u003d - 6 + 2 i 10 or x \u003d - 6 - 2 i 10,

x = - 3 5 + 1 5 i or x = - 3 5 - 1 5 i .

Answer: there are no real roots; the complex roots are: - 3 5 + 1 5 i , - 3 5 - 1 5 i .

IN school curriculum by default, there is no requirement to look for complex roots, therefore, if the discriminant is determined as negative during the solution, the answer is immediately recorded that there are no real roots.

Root formula for even second coefficients

The root formula x = - b ± D 2 a (D = b 2 − 4 a c) makes it possible to obtain another formula, more compact, allowing you to find solutions to quadratic equations with an even coefficient at x (or with a coefficient of the form 2 a n, for example, 2 3 or 14 ln 5 = 2 7 ln 5). Let us show how this formula is derived.

Suppose we are faced with the task of finding a solution to the quadratic equation a · x 2 + 2 · n · x + c = 0. We act according to the algorithm: we determine the discriminant D = (2 n) 2 − 4 a c = 4 n 2 − 4 a c = 4 (n 2 − a c) , and then use the root formula:

x \u003d - 2 n ± D 2 a, x \u003d - 2 n ± 4 n 2 - a c 2 a, x \u003d - 2 n ± 2 n 2 - a c 2 a, x = - n ± n 2 - a · c a .

Let the expression n 2 − a c be denoted as D 1 (sometimes it is denoted D "). Then the formula for the roots of the considered quadratic equation with the second coefficient 2 n will take the form:

x \u003d - n ± D 1 a, where D 1 \u003d 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 a quarter of the discriminant. Obviously, the sign of D 1 is the same as the sign of D, which means that the sign of D 1 can also serve as an indicator of the presence or absence of the roots of a quadratic equation.

Definition 11

Thus, to find a solution to a quadratic equation with a second coefficient of 2 n, it is necessary:

• find D 1 = n 2 − a c ;
• at D 1< 0 сделать вывод, что действительных корней нет;
• for D 1 = 0, determine the only root of the equation by the formula x = - n a ;
• for D 1 > 0, determine two real roots using the formula x = - n ± D 1 a.

Example 9

It is necessary to solve the quadratic equation 5 · x 2 − 6 · x − 32 = 0.

Solution

The second coefficient of the given equation can be represented as 2 · (− 3) . Then we rewrite the given quadratic equation as 5 · x 2 + 2 · (− 3) · x − 32 = 0 , where a = 5 , n = − 3 and c = − 32 .

Let's calculate the fourth part of the discriminant: D 1 = n 2 − a c = (− 3) 2 − 5 (− 32) = 9 + 160 = 169 . The resulting value is positive, which means that the equation has two real roots. We define them by the corresponding formula of the roots:

x = - n ± D 1 a , x = - - 3 ± 169 5 , x = 3 ± 13 5 ,

x = 3 + 13 5 or x = 3 - 13 5

x = 3 1 5 or x = - 2

It would be possible to perform calculations using the usual formula for the roots of a quadratic equation, but in this case the solution would be more cumbersome.

Answer: x = 3 1 5 or x = - 2 .

Simplification of the form of quadratic equations

Sometimes it is possible to optimize the form of the original equation, which will simplify the process of calculating the roots.

For example, the quadratic equation 12 x 2 - 4 x - 7 \u003d 0 is clearly more convenient for solving than 1200 x 2 - 400 x - 700 \u003d 0.

More often, the simplification of the form of a quadratic equation is performed by multiplying or dividing its both parts by a certain number. For example, above we showed a simplified representation of the equation 1200 x 2 - 400 x - 700 = 0, obtained by dividing both of its parts by 100.

Such a transformation is possible when the coefficients of the quadratic equation are not mutually prime numbers. Then it is common to divide both sides of the equation by the largest common divisor absolute values ​​of its coefficients.

As an example, we use the quadratic equation 12 x 2 − 42 x + 48 = 0. Let's define the gcd of the absolute values ​​of its coefficients: gcd (12 , 42 , 48) = gcd(gcd (12 , 42) , 48) = gcd (6 , 48) = 6 . Let's divide both parts of the original quadratic equation by 6 and get the equivalent quadratic equation 2 · x 2 − 7 · x + 8 = 0 .

By multiplying both sides of the quadratic equation, fractional coefficients are usually eliminated. In this case, multiply by the least common multiple of the denominators of its coefficients. For example, if each part of the quadratic equation 1 6 x 2 + 2 3 x - 3 \u003d 0 is multiplied with LCM (6, 3, 1) \u003d 6, then it will be written in more simple form x 2 + 4 x - 18 = 0 .

Finally, we note that almost always get rid of the minus at the first coefficient of the quadratic equation, changing the signs of each term of the equation, which is achieved by multiplying (or dividing) both parts by − 1. For example, from the quadratic equation - 2 x 2 - 3 x + 7 \u003d 0, you can go to its simplified version 2 x 2 + 3 x - 7 \u003d 0.

Relationship between roots and coefficients

The already known formula for the roots of quadratic equations x = - b ± D 2 · a expresses the roots of the equation in terms of its numerical coefficients. Based on this formula, we have the opportunity to set other dependencies between the roots and coefficients.

The most famous and applicable are the formulas of the Vieta theorem:

x 1 + x 2 \u003d - b a and x 2 \u003d c a.

In particular, for the given quadratic equation, the sum of the roots is the second coefficient with the opposite sign, and the product of the roots is equal to the free term. For example, by the form of the quadratic equation 3 · x 2 − 7 · x + 22 \u003d 0, it is possible to immediately determine that the sum of its roots is 7 3, and the product of the roots is 22 3.

You can also find a number of other relationships between the roots and coefficients of a quadratic equation. For example, the sum of the squares of the roots of a quadratic equation can be expressed in terms of coefficients:

x 1 2 + x 2 2 = (x 1 + x 2) 2 - 2 x 1 x 2 = - b a 2 - 2 c a = b 2 a 2 - 2 c a = b 2 - 2 a c a 2.

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This topic may seem difficult at first due to the many simple formulas. Not only do the quadratic equations themselves have long entries, but the roots are also found through the discriminant. There are three new formulas in total. Not very easy to remember. This is possible only after the frequent solution of such equations. Then all the formulas will be remembered by themselves.

General view of the quadratic equation

Here their explicit notation is proposed, when the largest degree is written first, and then - in descending order. Often there are situations when the terms stand apart. Then it is better to rewrite the equation in descending order of the degree of the variable.

Let us introduce notation. They are presented in the table below.

If we accept these notations, all quadratic equations are reduced to the following notation.

Moreover, the coefficient a ≠ 0. Let this formula be denoted by number one.

When the equation is given, it is not clear how many roots will be in the answer. Because one of three options is always possible:

• the solution will have two roots;
• the answer will be one number;
• The equation has no roots at all.

And while the decision is not brought to the end, it is difficult to understand which of the options will fall out in a particular case.

Types of records of quadratic equations

Tasks may have different entries. They will not always look like the general formula of a quadratic equation. Sometimes it will lack some terms. What was written above is the complete equation. If you remove the second or third term in it, you get something else. These records are also called quadratic equations, only incomplete.

Moreover, only the terms for which the coefficients "b" and "c" can disappear. The number "a" cannot be equal to zero under any circumstances. Because in this case the formula turns into a linear equation. The formulas for the incomplete form of the equations will be as follows:

So, there are only two types, in addition to complete ones, there are also incomplete quadratic equations. Let the first formula be number two, and the second number three.

The discriminant and the dependence of the number of roots on its value

This number must be known in order to calculate the roots of the equation. It can always be calculated, no matter what the formula of the quadratic equation is. In order to calculate the discriminant, you need to use the equality written below, which will have the number four.

After substituting the values ​​of the coefficients into this formula, you can get numbers with different signs. If the answer is yes, then the answer to the equation will be two different roots. With a negative number, the roots of the quadratic equation will be absent. If it is equal to zero, the answer will be one.

How is a complete quadratic equation solved?

In fact, consideration of this issue has already begun. Because first you need to find the discriminant. After it is clarified that there are roots of the quadratic equation, and their number is known, you need to use the formulas for the variables. If there are two roots, then you need to apply such a formula.

Since it contains the “±” sign, there will be two values. The expression under the square root sign is the discriminant. Therefore, the formula can be rewritten in a different way.

Formula five. From the same record it can be seen that if the discriminant is zero, then both roots will take the same values.

If the solution of quadratic equations has not yet been worked out, then it is better to write down the values ​​of all coefficients before applying the discriminant and variable formulas. Later this moment will not cause difficulties. But at the very beginning there is confusion.

How is an incomplete quadratic equation solved?

Everything is much simpler here. Even there is no need for additional formulas. And you won't need those that have already been written for the discriminant and the unknown.

First, consider the incomplete equation number two. In this equality, it is supposed to take the unknown value out of the bracket and solve the linear equation, which will remain in the brackets. The answer will have two roots. The first one is necessarily equal to zero, because there is a factor consisting of the variable itself. The second is obtained by solving a linear equation.

The incomplete equation at number three is solved by transferring the number from the left side of the equation to the right. Then you need to divide by the coefficient in front of the unknown. It remains only to extract the square root and do not forget to write it down twice with opposite signs.

The following are some actions that help you learn how to solve all kinds of equalities that turn into quadratic equations. They will help the student to avoid mistakes due to inattention. These shortcomings are the cause of poor grades when studying the extensive topic "Quadric Equations (Grade 8)". Subsequently, these actions will not need to be constantly performed. Because there will be a stable habit.

• First you need to write the equation in standard form. That is, first the term with the largest degree of the variable, and then - without the degree and the last - just a number.

• If a minus appears before the coefficient "a", then it can complicate the work for a beginner to study quadratic equations. It's better to get rid of it. For this purpose, all equality must be multiplied by "-1". This means that all terms will change sign to the opposite.

• In the same way, it is recommended to get rid of fractions. Simply multiply the equation by the appropriate factor so that the denominators cancel out.

Examples

It is required to solve the following quadratic equations:

x 2 - 7x \u003d 0;

15 - 2x - x 2 \u003d 0;

x 2 + 8 + 3x = 0;

12x + x 2 + 36 = 0;

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

The first equation: x 2 - 7x \u003d 0. It is incomplete, therefore it is solved as described for formula number two.

After bracketing, it turns out: x (x - 7) \u003d 0.

The first root takes the value: x 1 = 0. The second will be found from linear equation: x - 7 = 0. It is easy to see that x 2 = 7.

Second equation: 5x2 + 30 = 0. Again incomplete. Only it is solved as described for the third formula.

After transferring 30 to the right side of the equation: 5x 2 = 30. Now you need to divide by 5. It turns out: x 2 = 6. The answers will be numbers: x 1 = √6, x 2 = - √6.

Third equation: 15 - 2x - x 2 \u003d 0. Here and below, the solution of quadratic equations will begin by rewriting them in standard view: - x 2 - 2x + 15 = 0. Now it's time to use the second useful advice and multiply everything by minus one. It turns out x 2 + 2x - 15 \u003d 0. According to the fourth formula, you need to calculate the discriminant: D \u003d 2 2 - 4 * (- 15) \u003d 4 + 60 \u003d 64. It is a positive number. From what was said above, it turns out that the equation has two roots. They need to be calculated according to the fifth formula. According to it, it turns out that x \u003d (-2 ± √64) / 2 \u003d (-2 ± 8) / 2. Then x 1 \u003d 3, x 2 \u003d - 5.

The fourth equation x 2 + 8 + 3x \u003d 0 is converted to this: x 2 + 3x + 8 \u003d 0. Its discriminant is equal to this value: -23. Since this number is negative, the answer to this task will be the following entry: "There are no roots."

The fifth equation 12x + x 2 + 36 = 0 should be rewritten as follows: x 2 + 12x + 36 = 0. After applying the formula for the discriminant, the number zero is obtained. This means that it will have one root, namely: x \u003d -12 / (2 * 1) \u003d -6.

The sixth equation (x + 1) 2 + x + 1 = (x + 1) (x + 2) requires transformations, which consist in the fact that you need to bring like terms, before opening the brackets. In place of the first one there will be such an expression: x 2 + 2x + 1. After equality, this entry will appear: x 2 + 3x + 2. After similar terms are counted, the equation will take the form: x 2 - x \u003d 0. It has become incomplete . Similar to it has already been considered a little higher. The roots of this will be the numbers 0 and 1.

With this math program you can solve quadratic equation.

The program not only gives the answer to the problem, but also displays the solution process in two ways:
- using the discriminant
- using the Vieta theorem (if possible).

Moreover, the answer is displayed exact, not approximate.
For example, for the equation \(81x^2-16x-1=0\), the answer is displayed in this form:

$$ x_1 = \frac(8+\sqrt(145))(81), \quad x_2 = \frac(8-\sqrt(145))(81) $$ instead of this: \(x_1 = 0.247; \quad x_2 = -0.05 \)

This program can be useful for high school students general education schools in preparation for control work and exams, when testing knowledge before the exam, parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get it done as soon as possible? homework math or algebra? In this case, you can also use our programs with a detailed solution.

Thus, you can carry out your own training and/or training of their younger brothers or sisters, while the level of education in the field of tasks to be solved is increased.

If you are not familiar with the rules for entering a square polynomial, we recommend that you familiarize yourself with them.

Rules for entering a square polynomial

Any Latin letter can act as a variable.
For example: \(x, y, z, a, b, c, o, p, q \) etc.

Numbers can be entered as integers or fractions.
Moreover, fractional numbers can be entered not only in the form of a decimal, but also in the form of an ordinary fraction.

Rules for entering decimal fractions.
In decimal fractions, the fractional part from the integer can be separated by either a dot or a comma.
For example, you can enter decimals like this: 2.5x - 3.5x^2

Rules for entering ordinary fractions.
Only a whole number can act as the numerator, denominator and integer part of a fraction.

The denominator cannot be negative.

When entering a numerical fraction, the numerator is separated from the denominator by a division sign: /
The integer part is separated from the fraction by an ampersand: &
Input: 3&1/3 - 5&6/5z +1/7z^2
Result: \(3\frac(1)(3) - 5\frac(6)(5) z + \frac(1)(7)z^2 \)

When entering an expression you can use brackets. In this case, when solving a quadratic equation, the introduced expression is first simplified.
For example: 1/2(y-1)(y+1)-(5y-10&1/2)

=0

Decide

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A bit of theory.

Quadratic equation and its roots. Incomplete quadratic equations

Each of the equations
\(-x^2+6x+1,4=0, \quad 8x^2-7x=0, \quad x^2-\frac(4)(9)=0 \)
has the form
\(ax^2+bx+c=0, \)
where x is a variable, a, b and c are numbers.
In the first equation a = -1, b = 6 and c = 1.4, in the second a = 8, b = -7 and c = 0, in the third a = 1, b = 0 and c = 4/9. Such equations are called quadratic equations.

Definition.
quadratic equation an equation of the form ax 2 +bx+c=0 is called, where x is a variable, a, b and c are some numbers, and \(a \neq 0 \).

The numbers a, b and c are the coefficients of the quadratic equation. The number a is called the first coefficient, the number b is the second coefficient and the number c is the intercept.

In each of the equations of the form ax 2 +bx+c=0, where \(a \neq 0 \), the largest power of the variable x is a square. Hence the name: quadratic equation.

Note that a quadratic equation is also called an equation of the second degree, since its left side is a polynomial of the second degree.

A quadratic equation in which the coefficient at x 2 is 1 is called reduced quadratic equation. For example, the given quadratic equations are the equations
\(x^2-11x+30=0, \quad x^2-6x=0, \quad x^2-8=0 \)

If in the quadratic equation ax 2 +bx+c=0 at least one of the coefficients b or c is equal to zero, then such an equation is called incomplete quadratic equation. So, the equations -2x 2 +7=0, 3x 2 -10x=0, -4x 2 =0 are incomplete quadratic equations. In the first of them b=0, in the second c=0, in the third b=0 and c=0.

Incomplete quadratic equations are of three types:
1) ax 2 +c=0, where \(c \neq 0 \);
2) ax 2 +bx=0, where \(b \neq 0 \);
3) ax2=0.

Consider the solution of equations of each of these types.

To solve an incomplete quadratic equation of the form ax 2 +c=0 for \(c \neq 0 \), its free term is transferred to the right side and both parts of the equation are divided by a:
\(x^2 = -\frac(c)(a) \Rightarrow x_(1,2) = \pm \sqrt( -\frac(c)(a)) \)

Since \(c \neq 0 \), then \(-\frac(c)(a) \neq 0 \)

If \(-\frac(c)(a)>0 \), then the equation has two roots.

If \(-\frac(c)(a) To solve an incomplete quadratic equation of the form ax 2 +bx=0 for \(b \neq 0 \) factorize its left side and obtain the equation
\(x(ax+b)=0 \Rightarrow \left\( \begin(array)(l) x=0 \\ ax+b=0 \end(array) \right. \Rightarrow \left\( \begin (array)(l) x=0 \\ x=-\frac(b)(a) \end(array) \right. \)

Hence, an incomplete quadratic equation of the form ax 2 +bx=0 for \(b \neq 0 \) always has two roots.

An incomplete quadratic equation of the form ax 2 \u003d 0 is equivalent to the equation x 2 \u003d 0 and therefore has a single root 0.

The formula for the roots of a quadratic equation

Let us now consider how quadratic equations are solved in which both coefficients of the unknowns and the free term are nonzero.

We solve the quadratic equation in general form and as a result we obtain the formula of the roots. Then this formula can be applied to solve any quadratic equation.

Solve the quadratic equation ax 2 +bx+c=0

Dividing both its parts by a, we obtain the equivalent reduced quadratic equation
\(x^2+\frac(b)(a)x +\frac(c)(a)=0 \)

We transform this equation by highlighting the square of the binomial:
\(x^2+2x \cdot \frac(b)(2a)+\left(\frac(b)(2a)\right)^2- \left(\frac(b)(2a)\right)^ 2 + \frac(c)(a) = 0 \Rightarrow \)

\(x^2+2x \cdot \frac(b)(2a)+\left(\frac(b)(2a)\right)^2 = \left(\frac(b)(2a)\right)^ 2 - \frac(c)(a) \Rightarrow \) \(\left(x+\frac(b)(2a)\right)^2 = \frac(b^2)(4a^2) - \frac( c)(a) \Rightarrow \left(x+\frac(b)(2a)\right)^2 = \frac(b^2-4ac)(4a^2) \Rightarrow \) \(x+\frac(b )(2a) = \pm \sqrt( \frac(b^2-4ac)(4a^2) ) \Rightarrow x = -\frac(b)(2a) + \frac( \pm \sqrt(b^2 -4ac) )(2a) \Rightarrow \) \(x = \frac( -b \pm \sqrt(b^2-4ac) )(2a) \)

The root expression is called discriminant of a quadratic equation ax 2 +bx+c=0 (“discriminant” in Latin - distinguisher). It is denoted by the letter D, i.e.
\(D = b^2-4ac\)

Now, using the notation of the discriminant, we rewrite the formula for the roots of the quadratic equation:
\(x_(1,2) = \frac( -b \pm \sqrt(D) )(2a) \), where \(D= b^2-4ac \)

It's obvious that:
1) If D>0, then the quadratic equation has two roots.
2) If D=0, then the quadratic equation has one root \(x=-\frac(b)(2a)\).
3) If D Thus, depending on the value of the discriminant, the quadratic equation can have two roots (for D > 0), one root (for D = 0) or no roots (for D When solving a quadratic equation using this formula, it is advisable to do the following way:
1) calculate the discriminant and compare it with zero;
2) if the discriminant is positive or equal to zero, then use the root formula, if the discriminant is negative, then write down that there are no roots.

Vieta's theorem

The given quadratic equation ax 2 -7x+10=0 has roots 2 and 5. The sum of the roots is 7, and the product is 10. We see that the sum of the roots is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term. Any reduced quadratic equation that has roots has this property.

The sum of the roots of the given quadratic equation is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term.

Those. Vieta's theorem states that the roots x 1 and x 2 of the reduced quadratic equation x 2 +px+q=0 have the property:
\(\left\( \begin(array)(l) x_1+x_2=-p \\ x_1 \cdot x_2=q \end(array) \right. \)