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

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

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Quadratic equation, its types

Definition 1

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

Often, quadratic equations are also called equations of the second degree, since in essence a quadratic equation is algebraic equation 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. These 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 leading 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 use short form records like 6 x 2 − 2 x − 11 = 0, 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 leading coefficient is 1, and the second coefficient is − 1 .

Reduced and unreduced quadratic equations

Based on the value of the first coefficient, quadratic equations are divided into reduced and unreduced.

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.

Let's give 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 sides by the first coefficient (equivalent transformation). The transformed equation will have the same roots as the given unreduced equation or will also have no roots at all.

Consideration concrete example will allow us to clearly 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 sides 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's 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 precisely square, since at a = 0 it essentially transforms into a linear equation b x + c = 0.

In the case when 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- such a quadratic equation a x 2 + b x + c = 0, where at least one of the coefficients b And c(or both) is zero.

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

Let's discuss why the types of quadratic equations are given exactly these names.

When 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 obtained 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. Actually, this fact gave the name to this type of equation – incomplete.

For example, x 2 + 3 x + 4 = 0 and − 7 x 2 − 2 x + 1, 3 = 0 are complete quadratic equations; x 2 = 0, − 5 x 2 = 0; 11 x 2 + 2 = 0, − x 2 − 6 x = 0 – 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, this equation corresponds to the coefficients b = 0 and c = 0 ;
• a · x 2 + c = 0 at b = 0 ;
• a · x 2 + b · x = 0 at c = 0.

Let us consider sequentially the solution of each type of incomplete quadratic equation.

Solution of the equation a x 2 =0

As mentioned above, this equation corresponds to the coefficients b And c, equal to zero. Equation a x 2 = 0 can be converted into an equivalent equation x 2 = 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 x 2 = 0 this is zero because 0 2 = 0 . This equation has no other roots, which can be explained by the properties of the degree: for any number p, not equal to zero, the inequality is true p 2 > 0, from which it follows that when p ≠ 0 equality p 2 = 0 will never be achieved.

Definition 5

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

Example 2

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

Briefly, the solution is written as follows:

− 3 x 2 = 0, x 2 = 0, x = 0.

Solving the equation a x 2 + c = 0

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

• transfer c to the right hand side, which gives the equation a x 2 = − c;
• divide both sides of the equation by a, we end up with x = - c a .

Our transformations are equivalent; accordingly, the resulting equation is also equivalent to the original one, and this fact makes it possible to draw conclusions about the roots of the equation. From what the values ​​are a And c the value of the expression - c a depends: 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 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 the 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 = - c a will be the number - c a, since - c a 2 = - c a. It is not difficult 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 method of contradiction. To begin with, let us define the notations for the roots found above as x 1 And − x 1. Let us assume that the equation x 2 = - c a also has a root x 2, which is different from the roots x 1 And − x 1. We know that by substituting into the equation x its roots, we transform the equation into a fair numerical equality.

For x 1 And − x 1 we write: x 1 2 = - c a , and for x 2- x 2 2 = - c a . Based on the properties of numerical equalities, we subtract one correct equality term by term from another, which will give us: x 1 2 − x 2 2 = 0. We use the properties of operations with numbers 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 the above 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. An obvious contradiction arose, because at first it was agreed that the root of the equation x 2 different from x 1 And − x 1. So, we have proven that the equation has no roots other than x = - c a and x = - - c a.

Let us 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 have no roots at - c a< 0 ;
• will have two roots x = - c a and x = - - c a for - c a > 0.

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

Example 3

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

Solution

Let's move the free term to the right side of the equation, then the equation will take the form 9 x 2 = − 7.
Let us divide both sides of the resulting equation by 9 , we arrive at 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 have no roots.

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

Example 4

The equation needs to be solved − x 2 + 36 = 0.

Solution

Let's move 36 to the right side: − x 2 = − 36.
Let's divide both parts by − 1 , we get x 2 = 36. On the right side there is a positive number, from which we can conclude that x = 36 or x = - 36 .
Let's extract the root and write down the final result: incomplete quadratic equation − x 2 + 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 type of incomplete quadratic equations, when c = 0. To find a solution to an incomplete quadratic equation a x 2 + b x = 0, we will use the factorization method. Let's factorize the polynomial that 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 a set of equations x = 0 And a x + b = 0. 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 reinforce the material with an example.

Example 5

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

Solution

We'll take it out x outside the brackets we 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 write the solution to 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 for the roots of a quadratic equation

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

Definition 8

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

Writing x = - b ± D 2 · a essentially means that x 1 = - b + D 2 · a, x 2 = - b - D 2 · a.

It would be useful to understand how this formula was derived and how to apply it.

Derivation of the formula for the roots of a quadratic equation

Let us be 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 a number a, different from zero, we obtain the following quadratic equation: x 2 + b a · x + c a = 0 ;
• Let's select a complete 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 = 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 = b 2 4 · a 2 - c a = b 2 4 · a 2 - 4 · a · c 4 · a 2 = b 2 - 4 · a · c 4 · a 2 .

Thus, we arrive at the equation x + b 2 · a 2 = b 2 - 4 · a · c 4 · a 2 , equivalent to the original equation a x 2 + b x + c = 0.

We examined the solution of such equations in the previous paragraphs (solving 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:

• with b 2 - 4 a c 4 a 2< 0 уравнение не имеет действительных решений;
• when b 2 - 4 · a · c 4 · a 2 = 0 the equation is 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 following will be true: 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 roots of the equation x + b 2 · a 2 = b 2 - 4 · a · c 4 · a 2 (and therefore 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, (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 the name is given - the discriminant of the quadratic equation and the letter D is defined as its designation. Here you can write down the essence of the discriminant - based on its value and sign, they can conclude whether the quadratic equation will have real roots, and, if so, what is the number of 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 discriminant notation: x + b 2 · a 2 = D 4 · a 2 .

Let us formulate our conclusions again:

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 = - b 2 · a + D 4 · a 2 or x = - b 2 · a - D 4 · a 2. Based on the properties of radicals, these roots can be written in the form: x = - b 2 · a + D 2 · a or - b 2 · a - D 2 · a. And, when we open the modules and bring the fractions to a common denominator, we get: x = - b + D 2 · a, x = - b - D 2 · a.

So, the result of our reasoning was the derivation of the formula for the roots of a 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 to determine both real roots when the discriminant is greater than zero. When the discriminant is zero, applying both formulas will give the same root as the only solution to the quadratic equation. In the case where the discriminant is negative, if we try to use the formula for the root of a quadratic equation, we will be faced with the need to extract square root from a negative number, which will take us beyond the 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 this is generally done when it is necessary to find complex roots.

In the majority of cases, it usually means searching not for complex, but for real roots of a quadratic equation. Then it is optimal, before using the formulas for the roots of a quadratic equation, to first 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 discriminant value;
• at D< 0 сделать вывод об отсутствии у квадратного уравнения действительных корней;
• for D = 0, find the only root of the equation using the formula x = - b 2 · a ;
• for D > 0, determine two real roots of the quadratic equation using 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.

Let's look at examples.

Examples of solving quadratic equations

Let us give a solution to the examples for different meanings discriminant.

Example 6

We need to find the roots of the equation x 2 + 2 x − 6 = 0.

Solution

Let's write down the numerical coefficients of the quadratic equation: a = 1, b = 2 and c = − 6. Next we proceed according to the algorithm, i.e. Let's start calculating the discriminant, for which we will 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 get D > 0, which means that the original equation will have two real roots.
To find them, we use the root formula x = - b ± D 2 · a and, substituting the corresponding values, we get: x = - 2 ± 28 2 · 1. Let us simplify the resulting expression by taking the factor out of the root sign and then reducing 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

Need 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

The equation needs to be solved 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 calculated 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, performing actions with complex numbers:

x = - 6 ± - 4 2 5,

x = - 6 + 2 i 10 or x = - 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 as follows: - 3 5 + 1 5 · i, - 3 5 - 1 5 · i.

IN school curriculum There is no standard requirement to look for complex roots, therefore, if during the solution the discriminant is determined to be negative, the answer is immediately written down 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 one to find solutions to quadratic equations with an even coefficient for x (or with a coefficient of the form 2 · n, for example, 2 3 or 14 ln 5 = 2 7 ln 5). Let us show how this formula is derived.

Let us be faced with the task of finding a solution to the quadratic equation a · x 2 + 2 · n · x + c = 0 . We proceed 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 = - 2 n ± D 2 a, x = - 2 n ± 4 n 2 - a c 2 a, x = - 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 quadratic equation under consideration with the second coefficient 2 · n will take the form:

x = - n ± D 1 a, 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 a quarter of the discriminant. Obviously, the sign of D 1 is the same as the sign of D, which means the sign of D 1 can also serve as an indicator of the presence or absence of 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;
• when D 1 = 0, determine the only root of the equation using 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

We can represent the second coefficient of the given equation 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. Let us determine them using the corresponding root formula:

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 carry out 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 .

Simplifying 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 = 0 is clearly more convenient to solve than 1200 x 2 − 400 x − 700 = 0.

More often, simplification of the form of a quadratic equation is carried out by multiplying or dividing its both sides 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 sides by 100.

Such a transformation is possible when the coefficients of the quadratic equation are not mutually prime numbers. Then we usually 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 us determine the GCD of the absolute values ​​of its coefficients: GCD (12, 42, 48) = GCD(GCD (12, 42), 48) = GCD (6, 48) = 6. Let us divide both sides of the original quadratic equation by 6 and obtain the equivalent quadratic equation 2 x 2 − 7 x + 8 = 0.

By multiplying both sides of a quadratic equation, you usually get rid of fractional coefficients. In this case, they 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 = 0 is multiplied with LCM (6, 3, 1) = 6, then it will become written in more in simple form x 2 + 4 x − 18 = 0 .

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

Relationship between roots and coefficients

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

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

x 1 + x 2 = - b a and x 2 = 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 looking at the form of the quadratic equation 3 x 2 − 7 x + 22 = 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 connections 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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Solving equations in mathematics occupies a special place. This process is preceded by many hours of studying theory, during which the student learns how to solve equations, determine their type, and brings the skill to complete automation. However, the search for roots does not always make sense, since they may simply not exist. There are special moves finding roots. In this article we will analyze the main functions, their domains of definition, as well as cases when their roots are missing.

Which equation has no roots?

An equation has no roots if there are no real arguments x for which the equation is identically true. For a non-specialist, this formulation, like most mathematical theorems and formulas, looks very vague and abstract, but this is in theory. In practice, everything becomes extremely simple. For example: the equation 0 * x = -53 has no solution, since there is no number x whose product with zero would give something other than zero.

Now we will look at the most basic types of equations.

1. Linear equation

An equation is called linear if its right and left sides are represented as linear functions: ax + b = cx + d or in generalized form kx + b = 0. Where a, b, c, d are known numbers, and x is an unknown quantity . Which equation has no roots? Examples of linear equations are presented in the illustration below.

Basically, linear equations are solved by simply transferring the number part to one part and the contents of x to another. The result is an equation of the form mx = n, where m and n are numbers, and x is an unknown. To find x, just divide both sides by m. Then x = n/m. Most linear equations have only one root, but there are cases when there are either infinitely many roots or no roots at all. When m = 0 and n = 0, the equation takes the form 0 * x = 0. The solution to such an equation will be absolutely any number.

However, what equation has no roots?

For m = 0 and n = 0, the equation has no roots in the set of real numbers. 0 * x = -1; 0 * x = 200 - these equations have no roots.

2. Quadratic equation

A quadratic equation is an equation of the form ax 2 + bx + c = 0 for a = 0. The most common solution is through the discriminant. The formula for finding the discriminant of a quadratic equation is: D = b 2 - 4 * a * c. Next there are two roots x 1.2 = (-b ± √D) / 2 * a.

For D > 0 the equation has two roots, for D = 0 it has one root. But what quadratic equation has no roots? The easiest way to observe the number of roots of a quadratic equation is by graphing the function, which is a parabola. For a > 0 the branches are directed upward, for a< 0 ветви опущены вниз. Если дискриминант отрицателен, такое квадратное уравнение не имеет корней на множестве действительных чисел.

You can also visually determine the number of roots without calculating the discriminant. To do this, you need to find the vertex of the parabola and determine in which direction the branches are directed. The x coordinate of the vertex can be determined using the formula: x 0 = -b / 2a. In this case, the y coordinate of the vertex is found by simply substituting the x 0 value into the original equation.

The quadratic equation x 2 - 8x + 72 = 0 has no roots, since it has a negative discriminant D = (-8) 2 - 4 * 1 * 72 = -224. This means that the parabola does not touch the x-axis and the function never takes the value 0, therefore the equation has no real roots.

3. Trigonometric equations

Trigonometric functions are considered on a trigonometric circle, but can also be represented in a Cartesian coordinate system. In this article we will look at two basic trigonometric functions and their equations: sinx and cosx. Since these functions form trigonometric circle with radius 1, |sinx| and |cosx| cannot be more than 1. So, what sinx equation has no roots? Consider the graph of the sinx function shown in the picture below.

We see that the function is symmetric and has a repetition period of 2pi. Based on this, we can say that the maximum value of this function can be 1, and the minimum -1. For example, the expression cosx = 5 will not have roots, since its absolute value is greater than one.

This is the simplest example of trigonometric equations. In fact, solving them can take many pages, at the end of which you realize that you used the wrong formula and need to start all over again. Sometimes, even if you find the roots correctly, you may forget to take into account the restrictions on OD, which is why an extra root or interval appears in the answer, and the entire answer turns into an error. Therefore, strictly follow all the restrictions, because not all roots fit into the scope of the task.

4. Systems of equations

A system of equations is a set of equations joined by curly or square brackets. The curly brackets indicate that all equations are executed together. That is, if at least one of the equations does not have roots or contradicts another, the entire system has no solution. Square brackets indicate the word "or". This means that if at least one of the equations of the system has a solution, then the entire system has a solution.

The answer of the system c is the totality of all the roots of the individual equations. And systems with curly braces have only common roots. Systems of equations can include completely different functions, so such complexity does not allow us to immediately say which equation does not have roots.

Found in problem books and textbooks different types equations: those that have roots and those that do not. First of all, if you can’t find the roots, don’t think that they are not there at all. Perhaps you made a mistake somewhere, then you just need to carefully double-check your decision.

We looked at the most basic equations and their types. Now you can tell which equation has no roots. In most cases this is not difficult to do. Achieving success in solving equations requires only attention and concentration. Practice more, it will help you navigate the material much better and faster.

So, the equation has no roots if:

• V linear equation mx = n value m = 0 and n = 0;
• in a quadratic equation, if the discriminant is less than zero;
• V trigonometric equation of the form cosx = m / sinx = n, if |m| > 0, |n| > 0;
• in a system of equations with curly brackets, if at least one equation has no roots, and with square brackets, if all equations have no roots.

Quadratic equations. Discriminant. Solution, examples.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
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 the degree of 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.

What 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

Etc. 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 for given equation lead to standard view, i.e. to the form:

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.

It's 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 so. 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.

Have you figured it out? 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 it. 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.

So what of 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!
Doesn't work? That's it...
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. Oh well. 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 can't get by. 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. Did you understand that keyword Here - 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 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 common denominator, as described in the lesson "How to solve equations? Identical 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 it! Solving is a pleasure!

So, let's summarize the topic.

Practical advice:

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 out, 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 the solution different equations. Helps a lot!

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

IN modern society the ability to perform operations with equations containing a squared variable can be useful in many areas of activity and is widely used in practice in scientific and technical developments. Evidence of this can be found in the design of sea and river vessels, airplanes and rockets. Using such calculations, the trajectories of movement of a wide variety of bodies, including space objects, are determined. Examples with the solution of quadratic equations are used not only in economic forecasting, in the design and construction of buildings, but also in the most ordinary everyday circumstances. They may be needed on hiking trips, at sporting events, in stores when making purchases and in other very common situations.

Let's break the expression into its component factors

The degree of an equation is determined by the maximum value of the degree of the variable that the expression contains. If it is equal to 2, then such an equation is called quadratic.

If we speak in the language of formulas, then the indicated expressions, no matter how they look, can always be brought to the form when the left side of the expression consists of three terms. Among them: ax 2 (that is, a variable squared with its coefficient), bx (an unknown without a square with its coefficient) and c (a free component, that is, an ordinary number). All this on the right side is equal to 0. In the case when such a polynomial lacks one of its constituent terms, with the exception of ax 2, it is called an incomplete quadratic equation. Examples with the solution of such problems, the values ​​of the variables in which are easy to find, should be considered first.

If the expression looks in such a way that the expression on the right side has two terms, more precisely ax 2 and bx, the easiest way to find x is by placing the variable out of brackets. Now our equation will look like this: x(ax+b). Next, it becomes obvious that either x=0, or the problem comes down to finding a variable from the following expression: ax+b=0. This is dictated by one of the properties of multiplication. The rule states that the product of two factors results in 0 only if one of them is zero.

Example

x=0 or 8x - 3 = 0

As a result, we get two roots of the equation: 0 and 0.375.

Equations of this kind can describe the movement of bodies under the influence of gravity, which began to move from a certain point taken as the origin of coordinates. Here the mathematical notation takes the following form: y = v 0 t + gt 2 /2. By substituting the necessary values, equating the right side to 0 and finding possible unknowns, you can find out the time that passes from the moment the body rises to the moment it falls, as well as many other quantities. But we'll talk about this later.

Factoring an Expression

The rule described above makes it possible to solve these problems in more difficult cases. Let's look at examples of solving quadratic equations of this type.

X 2 - 33x + 200 = 0

This quadratic trinomial is complete. First, let's transform the expression and factor it. There are two of them: (x-8) and (x-25) = 0. As a result, we have two roots 8 and 25.

Examples with solving quadratic equations in grade 9 allow this method to find a variable in expressions not only of the second, but even of the third and fourth orders.

For example: 2x 3 + 2x 2 - 18x - 18 = 0. When factoring the right side into factors with a variable, there are three of them, that is, (x+1), (x-3) and (x+3).

As a result, it becomes obvious that this equation has three roots: -3; -1; 3.

Square Root

Another case incomplete equation the second order is an expression represented in the language of letters in such a way that the right side is constructed from the components ax 2 and c. Here, to obtain the value of the variable, the free term is transferred to right side, and after that the square root is taken from both sides of the equality. It should be noted that in in this case There are usually two roots of the equation. The only exceptions can be equalities that do not contain a term with at all, where the variable is equal to zero, as well as variants of expressions when the right side turns out to be negative. IN the latter case There are no solutions at all, since the above actions cannot be performed with roots. Examples of solutions to quadratic equations of this type should be considered.

In this case, the roots of the equation will be the numbers -4 and 4.

Calculation of land area

The need for this kind of calculations appeared in ancient times, because the development of mathematics was largely in those distant times was due to the need to determine with the greatest accuracy the areas and perimeters of land plots.

We should also consider examples of solving quadratic equations based on problems of this kind.

So, let's say there is a rectangular plot of land, the length of which is 16 meters greater than the width. You should find the length, width and perimeter of the site if you know that its area is 612 m2.

To get started, let's first create the necessary equation. Let us denote by x the width of the area, then its length will be (x+16). From what has been written it follows that the area is determined by the expression x(x+16), which, according to the conditions of our problem, is 612. This means that x(x+16) = 612.

Solving complete quadratic equations, and this expression is exactly that, cannot be done in the same way. Why? Although the left side still contains two factors, their product does not equal 0 at all, so different methods are used here.

Discriminant

First of all, let's make the necessary transformations, then appearance of this expression will look like this: x 2 + 16x - 612 = 0. This means that we have received an expression in a form corresponding to the previously specified standard, where a=1, b=16, c=-612.

This could be an example of solving quadratic equations using a discriminant. Here necessary calculations are produced according to the scheme: D = b 2 - 4ac. This auxiliary quantity not only makes it possible to find the required quantities in a second-order equation, it determines the quantity possible options. If D>0, there are two of them; for D=0 there is one root. In case D<0, никаких шансов для решения у уравнения вообще не имеется.

About roots and their formula

In our case, the discriminant is equal to: 256 - 4(-612) = 2704. This suggests that our problem has an answer. If you know k, the solution of quadratic equations must be continued using the formula below. It allows you to calculate the roots.

This means that in the presented case: x 1 =18, x 2 =-34. The second option in this dilemma cannot be a solution, because the dimensions of the land plot cannot be measured in negative quantities, which means x (that is, the width of the plot) is 18 m. From here we calculate the length: 18+16=34, and the perimeter 2(34+ 18)=104(m2).

Examples and tasks

We continue our study of quadratic equations. Examples and detailed solutions of several of them will be given below.

1) 15x 2 + 20x + 5 = 12x 2 + 27x + 1

Let’s move everything to the left side of the equality, make a transformation, that is, we’ll get the type of equation that is usually called standard, and equate it to zero.

15x 2 + 20x + 5 - 12x 2 - 27x - 1 = 0

Adding similar ones, we determine the discriminant: D = 49 - 48 = 1. This means our equation will have two roots. Let's calculate them according to the above formula, which means that the first of them will be equal to 4/3, and the second to 1.

2) Now let's solve mysteries of a different kind.

Let's find out if there are any roots here x 2 - 4x + 5 = 1? To obtain a comprehensive answer, let’s reduce the polynomial to the corresponding usual form and calculate the discriminant. In the above example, it is not necessary to solve the quadratic equation, because this is not the essence of the problem at all. In this case, D = 16 - 20 = -4, which means there really are no roots.

Vieta's theorem

It is convenient to solve quadratic equations using the above formulas and the discriminant, when the square root is taken from the value of the latter. But this does not always happen. However, there are many ways to obtain the values ​​of variables in this case. Example: solving quadratic equations using Vieta's theorem. She is named after someone who lived in 16th-century France and made a brilliant career thanks to his mathematical talent and connections at court. His portrait can be seen in the article.

The pattern that the famous Frenchman noticed was as follows. He proved that the roots of the equation add up numerically to -p=b/a, and their product corresponds to q=c/a.

Now let's look at specific tasks.

3x 2 + 21x - 54 = 0

For simplicity, let's transform the expression:

x 2 + 7x - 18 = 0

Let's use Vieta's theorem, this will give us the following: the sum of the roots is -7, and their product is -18. From here we get that the roots of the equation are the numbers -9 and 2. After checking, we will make sure that these variable values ​​really fit into the expression.

Parabola graph and equation

The concepts of quadratic function and quadratic equations are closely related. Examples of this have already been given earlier. Now let's look at some mathematical riddles in a little more detail. Any equation of the described type can be represented visually. Such a relationship, drawn as a graph, is called a parabola. Its various types are presented in the figure below.

Any parabola has a vertex, that is, a point from which its branches emerge. If a>0, they go high to infinity, and when a<0, они рисуются вниз. Простейшим примером подобной зависимости является функция y = x 2 . В данном случае в уравнении x 2 =0 неизвестное может принимать только одно значение, то есть х=0, а значит существует только один корень. Это неудивительно, ведь здесь D=0, потому что a=1, b=0, c=0. Выходит формула корней (точнее одного корня) квадратного уравнения запишется так: x = -b/2a.

Visual representations of functions help solve any equations, including quadratic ones. This method is called graphical. And the value of the variable x is the abscissa coordinate at the points where the graph line intersects with 0x. The coordinates of the vertex can be found using the formula just given x 0 = -b/2a. And by substituting the resulting value into the original equation of the function, you can find out y 0, that is, the second coordinate of the vertex of the parabola, which belongs to the ordinate axis.

The intersection of the branches of a parabola with the abscissa axis

There are a lot of examples of solving quadratic equations, but there are also general patterns. Let's look at them. It is clear that the intersection of the graph with the 0x axis for a>0 is possible only if y 0 takes negative values. And for a<0 координата у 0 должна быть положительна. Для указанных вариантов D>0. Otherwise D<0. А когда D=0, вершина параболы расположена непосредственно на оси 0х.

From the graph of the parabola you can also determine the roots. The opposite is also true. That is, if it is not easy to obtain a visual representation of a quadratic function, you can equate the right side of the expression to 0 and solve the resulting equation. And knowing the points of intersection with the 0x axis, it is easier to construct a graph.

From history

Using equations containing a squared variable, in the old days they not only made mathematical calculations and determined the areas of geometric figures. The ancients needed such calculations for grand discoveries in the fields of physics and astronomy, as well as for making astrological forecasts.

As modern scientists suggest, the inhabitants of Babylon were among the first to solve quadratic equations. This happened four centuries before our era. Of course, their calculations were radically different from those currently accepted and turned out to be much more primitive. For example, Mesopotamian mathematicians had no idea about the existence of negative numbers. They were also unfamiliar with other subtleties that any modern schoolchild knows.

Perhaps even earlier than the scientists of Babylon, the sage from India Baudhayama began solving quadratic equations. This happened about eight centuries before the era of Christ. True, the second-order equations, the methods for solving which he gave, were the simplest. Besides him, Chinese mathematicians were also interested in similar questions in the old days. In Europe, quadratic equations began to be solved only at the beginning of the 13th century, but later they were used in their works by such great scientists as Newton, Descartes and many others.

Quadratic equation problems are studied both in the school curriculum and in universities. They mean equations of the form a*x^2 + b*x + c = 0, where x- variable, a, b, c – constants; a<>0 . The task is to find the roots of the equation.

Geometric meaning of quadratic equation

The graph of a function that is represented by a quadratic equation is a parabola. The solutions (roots) of a quadratic equation are the points of intersection of the parabola with the abscissa (x) axis. It follows that there are three possible cases:
1) the parabola has no points of intersection with the abscissa axis. This means that it is in the upper plane with branches up or the bottom with branches down. In such cases, the quadratic equation has no real roots (it has two complex roots).

2) the parabola has one point of intersection with the Ox axis. Such a point is called the vertex of the parabola, and the quadratic equation at it acquires its minimum or maximum value. In this case, the quadratic equation has one real root (or two identical roots).

3) The last case is more interesting in practice - there are two points of intersection of the parabola with the abscissa axis. This means that there are two real roots of the equation.

Based on the analysis of the coefficients of the powers of the variables, interesting conclusions can be drawn about the placement of the parabola.

1) If coefficient a is greater than zero, then the parabola’s branches are directed upward, if negative, then the parabola’s branches are directed downward.

2) If the coefficient b is greater than zero, then the vertex of the parabola lies in the left half-plane, if it takes a negative value, then in the right half-plane.

Derivation of the formula for solving a quadratic equation

Let's transfer the constant from the quadratic equation

for the equal sign, we get the expression

Multiply both sides by 4a

To get a complete square on the left, add b^2 on both sides and carry out the transformation

From here we find

Formula for the discriminant and roots of a quadratic equation

The discriminant is the value of the radical expression. If it is positive, then the equation has two real roots, calculated by the formula When the discriminant is zero, the quadratic equation has one solution (two coinciding roots), which can be easily obtained from the above formula for D=0. When the discriminant is negative, the equation has no real roots. However, solutions to the quadratic equation are found in the complex plane, and their value is calculated using the formula

Vieta's theorem

Let's consider two roots of a quadratic equation and construct a quadratic equation on their basis. Vieta's theorem itself easily follows from the notation: if we have a quadratic equation of the form then the sum of its roots is equal to the coefficient p taken with the opposite sign, and the product of the roots of the equation is equal to the free term q. The formula for the above will look like If in a classical equation the constant a is nonzero, then you need to divide the entire equation by it, and then apply Vieta’s theorem.

Factoring quadratic equation schedule

Let the task be set: factor a quadratic equation. To do this, we first solve the equation (find the roots). Next, we substitute the found roots into the expansion formula for the quadratic equation. This will solve the problem.

Quadratic equation problems

Task 1. Find the roots of a quadratic equation

x^2-26x+120=0 .

Solution: Write down the coefficients and substitute them into the discriminant formula

The root of this value is 14, it is easy to find with a calculator, or remember with frequent use, however, for convenience, at the end of the article I will give you a list of squares of numbers that can often be encountered in such problems.
We substitute the found value into the root formula

and we get

Task 2. Solve the equation

2x 2 +x-3=0.

Solution: We have a complete quadratic equation, write out the coefficients and find the discriminant


Using known formulas we find the roots of the quadratic equation

Task 3. Solve the equation

9x 2 -12x+4=0.

Solution: We have a complete quadratic equation. Determining the discriminant

We got a case where the roots coincide. Find the values ​​of the roots using the formula

Task 4. Solve the equation

x^2+x-6=0 .

Solution: In cases where there are small coefficients for x, it is advisable to apply Vieta’s theorem. By its condition we obtain two equations

From the second condition we find that the product must be equal to -6. This means that one of the roots is negative. We have the following possible pair of solutions (-3;2), (3;-2) . Taking into account the first condition, we reject the second pair of solutions.
The roots of the equation are equal

Problem 5. Find the lengths of the sides of a rectangle if its perimeter is 18 cm and its area is 77 cm 2.

Solution: Half the perimeter of a rectangle is equal to the sum of its adjacent sides. Let's denote x as the larger side, then 18-x is its smaller side. The area of ​​the rectangle is equal to the product of these lengths:
x(18-x)=77;
or
x 2 -18x+77=0.
Let's find the discriminant of the equation

Calculating the roots of the equation

If x=11, That 18's=7 , the opposite is also true (if x=7, then 21's=9).

Problem 6. Factor the quadratic equation 10x 2 -11x+3=0.

Solution: Let's calculate the roots of the equation, to do this we find the discriminant

We substitute the found value into the root formula and calculate

We apply the formula for decomposing a quadratic equation by roots

Opening the brackets we obtain an identity.

Quadratic equation with parameter

Example 1. At what parameter values A , does the equation (a-3)x 2 + (3-a)x-1/4=0 have one root?

Solution: By direct substitution of the value a=3 we see that it has no solution. Next, we will use the fact that with a zero discriminant the equation has one root of multiplicity 2. Let's write out the discriminant

Let's simplify it and equate it to zero

We have obtained a quadratic equation with respect to the parameter a, the solution of which can be easily obtained using Vieta’s theorem. The sum of the roots is 7, and their product is 12. By simple search we establish that the numbers 3,4 will be the roots of the equation. Since we already rejected the solution a=3 at the beginning of the calculations, the only correct one will be - a=4. Thus, when a=4 the equation has one root.

Example 2. At what parameter values A , equation a(a+3)x^2+(2a+6)x-3a-9=0 has more than one root?

Solution: Let's first consider the singular points, they will be the values ​​a=0 and a=-3. When a=0, the equation will be simplified to the form 6x-9=0; x=3/2 and there will be one root. For a= -3 we obtain the identity 0=0.
Let's calculate the discriminant

and find the value of a at which it is positive

From the first condition we get a>3. For the second, we find the discriminant and roots of the equation


Let us determine the intervals where the function takes positive values. By substituting the point a=0 we get 3>0 . So, outside the interval (-3;1/3) the function is negative. Don't forget the point a=0, which should be excluded because the original equation has one root in it.
As a result, we obtain two intervals that satisfy the conditions of the problem

There will be many similar tasks in practice, try to figure out the tasks yourself and do not forget to take into account the conditions that are mutually exclusive. Study well the formulas for solving quadratic equations; they are often needed in calculations in various problems and sciences.