Quadratic equation with large numbers. Properties of the coefficients of a quadratic equation. Quadratic equations. average level

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 the quadratic equation, set the accompanying terms, analyze the scheme for solving incomplete and complete equations, we will get acquainted with the formula of roots and the discriminant, we will establish connections between the roots and coefficients, and of course we will give a visual solution of practical examples.

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 the shorthand form is used 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 of a specific 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 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 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) 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 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 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 is a positive number, from which we can conclude 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;
• select the full square on the left side of the resulting equation:
  x 2 + ba x + ca = x 2 + 2 b 2 a x + b 2 a 2 - b 2 a 2 + ca = = x + b 2 a 2 - b 2 a 2 + ca
  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 the square root of negative number that 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

We present the solution of examples for various values ​​of the 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 the school curriculum, as a standard, there is no requirement to look for complex roots, therefore, if the discriminant is defined 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 \u003d - n ± n 2 - a · ca.

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 relatively prime numbers. Then, usually, both parts of the equation are divided by the greatest common divisor of the 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 a simpler 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 = - ba 2 - 2 ca = b 2 a 2 - 2 ca = b 2 - 2 a ca 2.

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Quadratic equations. General information.

V quadratic equation there must be an x ​​in the square (that's why it is called

"square"). In addition to it, in the equation there may be (or may not be!) Just x (to the first degree) and

just a number (free member). And there should not be x's in a degree greater than two.

Algebraic equation of general form.

where x is a free variable, a, b, c are coefficients, and a≠0 .

for instance:

Expression called square trinomial.

The elements of a quadratic equation have their own names:

called the first or senior coefficient,

is called the second or coefficient at ,

is called a free member.

Complete quadratic equation.

These quadratic equations have the full set of terms on the left. x squared

coefficient a, x to the first power with coefficient b and free memberWith. V all coefficients

must be different from zero.

Incomplete is a quadratic equation in which at least one of the coefficients, except for

senior (either the second coefficient or the free term) is equal to zero.

Let's pretend that b\u003d 0, - x will disappear in the first degree. It turns out, for example:

2x 2 -6x=0,

Etc. And if both coefficients b and c are equal to zero, then it is even simpler, For example:

2x 2 \u003d 0,

Note that x squared is present in all equations.

Why a can't be zero? Then the x squared disappears and the equation becomes linear .

And it's done differently...

The graph of a quadratic function is a parabola. The solutions (roots) of a quadratic equation are the points of intersection of the parabola with the abscissa axis. If the parabola described by the quadratic function does not intersect the x-axis, the equation has no real roots. If the parabola intersects the x-axis at one point (the apex of the parabola), the equation has one real root (the equation is also said to have two coinciding roots). If the parabola intersects the x-axis at two points, the equation has two real roots.

If the coefficient a If it is positive, the parabola branches are directed upwards; if it is negative, the parabola branches are directed downwards. If the coefficient b is positive, then the vertex of the parabola lies in the left half-plane, if negative - in the right half-plane.

Derivation of a formula for solving a quadratic equation

The formula for solving a quadratic equation can be obtained as follows

a x 2 + b x + c = 0
a x 2 + b x=- c

Multiply the equation by 4 a

4a 2 x 2 + 4 ab x=-4 ac
4a 2 x 2 + 4 ab x + b 2 = -4ac + b 2
(2a x + b) 2 = b 2 -4ac
2a x + b= ±$\sqrt(b^2-4 a c)$

Finding the roots of a quadratic equation

A quadratic equation with real coefficients can have from 0 to 2 real roots, depending on the value of the discriminant D = b 2 − 4ac:

• for D > 0 there are two roots, and they are calculated by the formula
• for D = 0, there is one root (two equal or coinciding roots), multiplicity 2:

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

First, we will discuss what a quadratic equation is, how it is written in general form, and give related definitions. After that, using examples, we will analyze in detail how incomplete quadratic equations are solved. Next, we move on to solving complete equations, get the formula for the roots, get acquainted with the discriminant of a quadratic equation and consider solutions to typical examples. Finally, we trace the connections between roots and coefficients.

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What is a quadratic equation? Their types

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

Definition and examples of quadratic equations

Definition.

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

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

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

Definition.

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

For example, let's take a quadratic equation of the form 5 x 2 −2 x−3=0 , here the leading coefficient is 5 , the second coefficient is −2 , and the free term is −3 . Note that when the coefficients b and/or c are negative, as in the example just given, the short form of the quadratic equation of the form 5 x 2 −2 x−3=0 is used, not 5 x 2 +(−2 )x+(−3)=0 .

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

Reduced and non-reduced quadratic equations

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

Definition.

A quadratic equation in which the leading coefficient is 1 is called reduced quadratic equation. Otherwise, the quadratic equation is unreduced.

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

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

Let's take an example of how the transition from an unreduced quadratic equation to a reduced one is performed.

Example.

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

Solution.

It is enough for us to perform the division of both parts of the original equation by the leading coefficient 3, it is non-zero, so we can perform this action. We have (3 x 2 +12 x−7):3=0:3 , which is the same as (3 x 2):3+(12 x):3−7:3=0 , and so on (3:3) x 2 +(12:3) x−7:3=0 , whence . So we got the reduced quadratic equation, which is equivalent to the original one.

Answer:

Complete and incomplete quadratic equations

There is a condition a≠0 in the definition of a quadratic equation. This condition is necessary in order for the equation a x 2 +b x+c=0 to be exactly square, since with a=0 it actually becomes a linear equation of the form b x+c=0 .

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

Definition.

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

In turn

Definition.

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

These names are not given by chance. This will become clear from the following discussion.

If the coefficient b is equal to zero, then the quadratic equation takes the form a x 2 +0 x+c=0 , and it is equivalent to the equation a x 2 +c=0 . If c=0 , that is, the quadratic equation has the form a x 2 +b x+0=0 , then it can be rewritten as a x 2 +b x=0 . And with b=0 and c=0 we get the quadratic equation a·x 2 =0. The resulting equations differ from the 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. Hence their name - incomplete quadratic equations.

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

Solving incomplete quadratic equations

It follows from the information of the previous paragraph that there is three kinds of incomplete quadratic equations:

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

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

a x 2 \u003d 0

Let's start by solving incomplete quadratic equations in which the coefficients b and c are equal to zero, that is, with equations of the form a x 2 =0. The equation a·x 2 =0 is equivalent to the equation x 2 =0, which is obtained from the original by dividing its both parts by a non-zero number a. Obviously, the root of the equation x 2 \u003d 0 is zero, since 0 2 \u003d 0. This equation has no other roots, which is explained, indeed, for any non-zero number p, the inequality p 2 >0 takes place, which implies that for p≠0, the equality p 2 =0 is never achieved.

So, the incomplete quadratic equation a x 2 \u003d 0 has a single root x \u003d 0.

As an example, we give the solution of an incomplete quadratic equation −4·x 2 =0. It is equivalent to the equation x 2 \u003d 0, its only root is x \u003d 0, therefore, the original equation has a single root zero.

A short solution in this case can be issued as follows:
−4 x 2 \u003d 0,
x 2 \u003d 0,
x=0 .

a x 2 +c=0

Now consider how incomplete quadratic equations are solved, in which the coefficient b is equal to zero, and c≠0, that is, equations of the form a x 2 +c=0. We know that the transfer of a term from one side of the equation to the other with the opposite sign, as well as the division of both sides of the equation by a non-zero number, give an equivalent equation. Therefore, the following equivalent transformations of the incomplete quadratic equation a x 2 +c=0 can be carried out:

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

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

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

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

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

Let's summarize the information in this paragraph. The incomplete quadratic equation a x 2 +c=0 is equivalent to the equation , which

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

Consider examples of solving incomplete quadratic equations of the form a·x 2 +c=0 .

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

Let's solve one more incomplete quadratic equation −x 2 +9=0. We transfer the nine to the right side: -x 2 \u003d -9. Now we divide both parts by −1, we get x 2 =9. The right side contains a positive number, from which we conclude that or . After we write down the final answer: the incomplete quadratic equation −x 2 +9=0 has two roots x=3 or x=−3.

a x 2 +b x=0

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

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

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

Example.

Solve the equation.

Solution.

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

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

Answer:

x=0 , .

Discriminant, formula of the roots of a quadratic equation

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

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

Derivation of the formula of the roots of a quadratic equation

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

• We can divide both parts of this equation by a non-zero number a, as a result we get the reduced quadratic equation.
• Now select a full square on its left side: . After that, the equation will take the form .
• At this stage, it is possible to carry out the transfer of the last two terms to the right side with the opposite sign, we have .
• And let's also transform the expression on the right side: .

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

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

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

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

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

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

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

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

Algorithm for solving quadratic equations using root formulas

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

However, in a school algebra course, we usually talk not about complex, but about real roots of a quadratic equation. In this case, it is advisable to first find the discriminant before using the formulas for the roots of the quadratic equation, make sure that it is non-negative (otherwise, we can conclude that the equation has no real roots), and after that calculate the values ​​of the roots.

The above reasoning allows us to write algorithm for solving a quadratic equation. To solve the quadratic equation a x 2 + b x + c \u003d 0, you need:

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

Here we only note that if the discriminant is equal to zero, the formula can also be used, it will give the same value as .

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

Examples of solving quadratic equations

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

Example.

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

Solution.

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

Answer:

Let's move on to the next typical example.

Example.

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

Solution.

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

Answer:

x=3.5 .

It remains to consider the solution of quadratic equations with negative discriminant.

Example.

Solve the equation 5 y 2 +6 y+2=0 .

Solution.

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

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

Answer:

there are no real roots, the complex roots are: .

Once again, we note that if the discriminant of the quadratic equation is negative, then the school usually immediately writes down the answer, in which they indicate that there are no real roots, and they do not find complex roots.

Root formula for even second coefficients

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

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

Denote the expression n 2 − a c 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 takes the form , where D 1 =n 2 −a c .

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

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

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

Consider the solution of the example using the root formula obtained in this paragraph.

Example.

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

Solution.

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

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

Answer:

Simplification of the form of quadratic equations

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

Usually, a simplification of the form of a quadratic equation is achieved by multiplying or dividing both sides of it by some number. For example, in the previous paragraph, we managed to achieve a simplification of the equation 1100 x 2 −400 x −600=0 by dividing both sides by 100 .

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

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

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

Relationship between roots and coefficients of a quadratic equation

The formula for the roots of a quadratic equation expresses the roots of an equation in terms of its coefficients. Based on the formula of the roots, you can get other relationships between the roots and coefficients.

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

Using the already written formulas, you can get a number of other relationships between the roots and coefficients of the quadratic equation. For example, you can express the sum of the squares of the roots of a quadratic equation in terms of its coefficients: .

Bibliography.

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

Some problems in mathematics require the ability to calculate the value of the square root. These problems include solving second-order equations. In this article, we present an effective method for calculating square roots and use it when working with formulas for the roots of a quadratic equation.

What is a square root?

In mathematics, this concept corresponds to the symbol √. Historical data says that it began to be used for the first time around the first half of the 16th century in Germany (the first German work on algebra by Christoph Rudolf). Scientists believe that this symbol is a transformed Latin letter r (radix means "root" in Latin).

The root of any number is equal to such a value, the square of which corresponds to the root expression. In the language of mathematics, this definition will look like this: √x = y if y 2 = x.

root of positive number(x > 0) is also a positive number (y > 0), but if you take the root of a negative number (x< 0), то его результатом уже будет комплексное число, включающее мнимую единицу i.

Here are two simple examples:

√9 = 3 because 3 2 = 9; √(-9) = 3i since i 2 = -1.

Heron's iterative formula for finding the values ​​of the square roots

The above examples are very simple, and the calculation of the roots in them is not difficult. Difficulties begin to appear already when finding the root values ​​for any value that cannot be represented as a square of a natural number, for example √10, √11, √12, √13, not to mention the fact that in practice it is necessary to find roots for non-integer numbers: for example √(12.15), √(8.5) and so on.

In all the above cases, a special method for calculating the square root should be used. At present, several such methods are known: for example, expansion in a Taylor series, division by a column, and some others. Of all known methods, perhaps the most simple and effective is the use of Heron's iterative formula, which is also known as the Babylonian method for determining square roots (there is evidence that the ancient Babylonians used it in their practical calculations).

Let it be necessary to determine the value of √x. The formula for finding the square root is as follows:

a n+1 = 1/2(a n +x/a n), where lim n->∞ (a n) => x.

Let's decipher this mathematical notation. To calculate √x, you should take some number a 0 (it can be arbitrary, however, to quickly get the result, you should choose it so that (a 0) 2 is as close as possible to x. Then substitute it into the specified formula for calculating the square root and get a new the number a 1, which will already be closer to the desired value.After that, it is necessary to substitute a 1 into the expression and get a 2. This procedure should be repeated until the required accuracy is obtained.

An example of applying Heron's iterative formula

The algorithm described above for obtaining the square root of some given number may sound rather complicated and confusing for many, but in reality everything turns out to be much simpler, since this formula converges very quickly (especially if a good number a 0 is chosen).

Let's give a simple example: it is necessary to calculate √11. We choose a 0 \u003d 3, since 3 2 \u003d 9, which is closer to 11 than 4 2 \u003d 16. Substituting into the formula, we get:

a 1 \u003d 1/2 (3 + 11/3) \u003d 3.333333;

a 2 \u003d 1/2 (3.33333 + 11 / 3.33333) \u003d 3.316668;

a 3 \u003d 1/2 (3.316668 + 11 / 3.316668) \u003d 3.31662.

There is no point in continuing the calculations, since we have found that a 2 and a 3 begin to differ only in the 5th decimal place. Thus, it was enough to apply the formula only 2 times to calculate √11 with an accuracy of 0.0001.

At present, calculators and computers are widely used to calculate the roots, however, it is useful to remember the marked formula in order to be able to manually calculate their exact value.

Second order equations

Understanding what a square root is and the ability to calculate it is used when solving quadratic equations. These equations are equalities with one unknown, general form which is shown in the figure below.

Here c, b and a are some numbers, and a must not be equal to zero, and the values ​​of c and b can be completely arbitrary, including being equal to zero.

Any values ​​of x that satisfy the equality indicated in the figure are called its roots (this concept should not be confused with the square root √). Since the equation under consideration has the 2nd order (x 2), then there cannot be more roots for it than two numbers. We will consider later in the article how to find these roots.

Finding the roots of a quadratic equation (formula)

This method of solving the type of equalities under consideration is also called universal, or the method through the discriminant. It can be applied to any quadratic equations. The formula for the discriminant and roots of the quadratic equation is as follows:

It can be seen from it that the roots depend on the value of each of the three coefficients of the equation. Moreover, the calculation of x 1 differs from the calculation of x 2 only by the sign in front of the square root. The radical expression, which is equal to b 2 - 4ac, is nothing more than the discriminant of the considered equality. The discriminant in the formula for the roots of a quadratic equation plays an important role because it determines the number and type of solutions. So, if it is zero, then there will be only one solution, if it is positive, then the equation has two real roots, and finally, a negative discriminant leads to two complex roots x 1 and x 2.

Vieta's theorem or some properties of the roots of second-order equations

At the end of the 16th century, one of the founders of modern algebra, a Frenchman, studying second-order equations, was able to obtain the properties of its roots. Mathematically, they can be written like this:

x 1 + x 2 = -b / a and x 1 * x 2 = c / a.

Both equalities can easily be obtained by everyone; for this, it is only necessary to perform the appropriate mathematical operations with the roots obtained through a formula with a discriminant.

The combination of these two expressions can rightfully be called the second formula of the roots of a quadratic equation, which makes it possible to guess its solutions without using the discriminant. It should be noted here that although both expressions are always valid, it is convenient to use them to solve an equation only if it can be factored.

The task of consolidating the acquired knowledge

We will solve a mathematical problem in which we will demonstrate all the techniques discussed in the article. The conditions of the problem are as follows: you need to find two numbers for which the product is -13, and the sum is 4.

This condition immediately reminds of Vieta's theorem, using the formulas for the sum of square roots and their product, we write:

x 1 + x 2 \u003d -b / a \u003d 4;

x 1 * x 2 \u003d c / a \u003d -13.

Assuming a = 1, then b = -4 and c = -13. These coefficients allow us to compose a second-order equation:

x 2 - 4x - 13 = 0.

We use the formula with the discriminant, we get the following roots:

x 1.2 = (4 ± √D)/2, D = 16 - 4 * 1 * (-13) = 68.

That is, the task was reduced to finding the number √68. Note that 68 = 4 * 17, then, using the square root property, we get: √68 = 2√17.

Now we use the considered square root formula: a 0 \u003d 4, then:

a 1 \u003d 1/2 (4 + 17/4) \u003d 4.125;

a 2 \u003d 1/2 (4.125 + 17 / 4.125) \u003d 4.1231.

There is no need to calculate a 3 because the found values ​​differ by only 0.02. Thus, √68 = 8.246. Substituting it into the formula for x 1,2, we get:

x 1 \u003d (4 + 8.246) / 2 \u003d 6.123 and x 2 \u003d (4 - 8.246) / 2 \u003d -2.123.

As you can see, the sum of the numbers found is really equal to 4, but if you find their product, then it will be equal to -12.999, which satisfies the condition of the problem with an accuracy of 0.001.