The formula when the discriminant is equal to a negative number. Discriminant: Examples of solving equations

Square equation is an equation that looks like aX 2 + DX + C \u003d 0. In it, the value a, B. and from any numbers but Not equally zero.

All square equations are divided into several species, namely:

Equations in which only one root.
-Evaluation with two different roots.
-Evaluation in which there are no roots at all.

This distinguishes linear equations in which the root is always united, from square. In order to understand how much the number of roots in the expression and need Discriminant square equation.

Let's say our equation AX 2 + DX + C \u003d 0. So Discriminant square equation -

D \u003d b 2 - 4 AC

And it needs to be remembered forever. With this equation, we determine the number of roots in the square equation. And we do this as follows:

When D is less than zero, there are no roots in the equation.
- When D is zero, there is only one root.
- When D is larger, respectively, in the two root equation.
Remember that the discriminant shows how many roots in the equation, without changing signs.

Consider for clarity:

It is necessary to find out what the number of roots in this square equation.

1) x 2 - 8x + 12 \u003d 0
2) 5x 2 + 3x + 7 \u003d 0
3) x 2 -6x + 9 \u003d 0

Enter the values \u200b\u200bin the first equation, we find the discriminant.
a \u003d 1, b \u003d -8, c \u003d 12
D \u003d (-8) 2 - 4 * 1 * 12 \u003d 64 - 48 \u003d 16
Discriminant with a plus sign, which means two roots in this equality.

Do the same with the second equation
a \u003d 1, b \u003d 3, c \u003d 7
D \u003d 3 2 - 4 * 5 * 7 \u003d 9 - 140 \u003d - 131
The value is minus, which means no roots in this equality.

The following equation is decomposable by analogy.
a \u003d 1, b \u003d -6, c \u003d 9
D \u003d (-6) 2 - 4 * 1 * 9 \u003d 36 - 36 \u003d 0
As a result, we have one root in the equation.

It is important that in each equation we discharged the coefficients. Of course, this is not a lot of a long process, but it helped us not to get confused and prevented the appearance of errors. If you often solve such equations, then the calculations can be made mentally and in advance to know how many roots in the equation.

Consider another example:

1) x 2 - 2x - 3 \u003d 0
2) 15 - 2x - x 2 \u003d 0
3) x 2 + 12x + 36 \u003d 0

Unlock first
a \u003d 1, b \u003d -2, c \u003d -3
D \u003d (- 2) 2 - 4 * 1 * (-3) \u003d 16, which is more zero, then two roots, bring them
x 1 \u003d 2+? 16/2 * 1 \u003d 3, x 2 \u003d 2-? 16/2 * 1 \u003d -1.

We declare second
a \u003d -1, b \u003d -2, c \u003d 15
D \u003d (-2) 2 - 4 * 4 * (-1) * 15 \u003d 64, which is more zero and also has two roots. Let's bring them:
x 1 \u003d 2+? 64/2 * (-1) \u003d -5, x 2 \u003d 2-? 64/2 * (- 1) \u003d 3.

Unlock the third
a \u003d 1, b \u003d 12, c \u003d 36
D \u003d 12 2 - 4 * 1 * 36 \u003d 0, which is zero and has one root
x \u003d -12 +? 0/2 * 1 \u003d -6.
It is not difficult to solve these equations.

If we are given an incomplete square equation. Such as

1x 2 + 9x \u003d 0
2x 2 - 16 \u003d 0

These equations differ from those that were higher, as it is not complete, there is no third value in it. But despite this it is easier than a complete square equation and does not need to search for a discriminant.

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Among the entire course of the school program, the algebra of one of the most voluminous topics is the topic of square equations. At the same time, under the square equation, the equation of the form Ax 2 + BX + C \u003d 0, where A ≠ 0 (reads: and multiply to X in square plus BE X plus CE is zero, where and unequal zero). In this case, the main place is occupied by the formulas of the discriminant of the square equation of the specified species, under which the expression is understood to determine the presence or absence of roots in the square equation, as well as their number (if available).

Formula (equation) of the discriminant of a square equation

The generally accepted formula of the discriminant of the square equation is as follows: D \u003d B 2 - 4AC. Calculating the discriminant according to the specified formula, you can not only determine the presence and number of roots in the square equation, but also choose the method of finding these roots, which exist somewhat depending on the type of square equation.

What does that mean discriminant is zero \\ the root formula of the square equation if the discriminant is zero

The discriminant, as follows from the formula, is indicated by the Latin letter D. In the case when the discriminant is zero, it should be concluded that the square equation of the form AX 2 + BX + C \u003d 0, where a ≠ 0, has only one root that is calculated by Simplified formula. This formula is applied only at zero discriminator and is as follows: x \u003d -b / 2a, where X is the root of the square equation, B and A - the corresponding variables of the square equation. To find the root of the square equation, the negative value of the variable B is to divide the double value of the variable a. The resulting expression will be solved by a square equation.

Solution of the square equation through discriminant

If, when calculating the discriminant according to the above formula, a positive value is obtained (D greater than zero), the square equation has two roots that are calculated according to the following formulas: x 1 \u003d (-b + Vd) / 2a, x 2 \u003d (-b - Vd) / 2A. Most often, the discriminant is not considered separately, and in the value of D, from which the root is extracted, the guided expression is simply substituted in the form of a discriminant formula. If the variable B has an even meaning, then to calculate the roots of the square equation of the form AX 2 + BX + C \u003d 0, where a ≠ 0 can also use the following formulas: x 1 \u003d (-k + V (K2 - AC)) / a , x 2 \u003d (-k + v (k2 - AC)) / A, where k \u003d b / 2.

In some cases, for the practical solution of square equations, the Vieta Theorem can be used, which states that for the amount of the roots of the square equation of the form x 2 + px + q \u003d 0, the value x 1 + x 2 \u003d -p will be true, and for the product of the roots of the specified equation - expression x 1 xx 2 \u003d Q.

Can the discriminant be less than zero

When calculating the discriminant value, you can encounter a situation that does not fall under any of the cases described - when the discriminant has a negative value (that is, less than zero). In this case, it is believed that the square equation of the AX 2 + BX + C \u003d 0 form, where a ≠ 0, the valid roots does not have, therefore, its solution will be limited to the discriminant calculation, and the above-mentioned formulas of the square equation in this case are not applied will be. At the same time, in response to the square equation, it is recorded that "the equation of valid roots does not have".

Explanatory video:

Discriminant, like square equations begin to study in the course of algebra in grade 8. It is possible to solve the square equation through the discriminant and using the Vieta theorem. The methodology for studying square equations, as well as the formulas of the discriminant, is unsuccessful to schoolchildren, as well as much in this education. Therefore, school years are passing, training in grade 9-11 replaces "Higher Education" and everyone is looking for - "How to solve a square equation?", "How to find the roots of the equation?", "How to find a discriminant?" and...

Formula discriminant

Discriminant D of the square equation A * X ^ 2 + BX + C \u003d 0 is d \u003d b ^ 2-4 * a * c.
Roots (solutions) of the square equation depend on the discriminant sign (D):
D\u003e 0 - the equation has 2 different valid roots;
D \u003d 0 - the equation has 1 root (2 coinciding root):
D.<0 – не имеет действительных корней (в школьной теории). В ВУЗах изучают комплексные числа и уже на множестве комплексных чисел уравнение с отрицательным дискриминантом имеет два комплексных корня.
The formula for calculating the discriminant is quite simple, so many sites offer an online discriminant calculator. We have not figured out this kind of scripts, so who knows how to implement it please write to the post office This email address is protected from spam bots. You must have JavaScript enabled to view. .

General formula for finding the roots of the square equation:

Roots equations find by formula
If the coefficient with a variable in the square is paired, it is advisable to calculate the discriminant, but the fourth part of it
In such cases, the roots of the equation are found by the formula

The second way of finding roots is the Vieta Theorem.

The theorem is formulated not only for square equations, but also for polynomials. You can read this in Wikipedia or other electronic resources. However, to simplify, consider it part of it, which concerns the above square equations, that is, the equations of the form (A \u003d 1)
The essence of the formulas of the wine is that the amount of the roots of the equation is equal to the coefficient with a variable taken with the opposite sign. The product of the roots of the equation is equal to a free member. The formulas of the Vieta Theorem has a record.
The output of the formula of the Vieta is quite simple. Cut the square equation through simple multipliers
As you can see, everything ingenious is simultaneously simple. Effectively use the wine formula when the root difference in the module or the difference of roots modules is 1, 2. For example, the following equations on the Vieta theorem have roots




Up to 4 equations, the analysis should look as follows. The product of the equation equation is 6, therefore, the roots may be values \u200b\u200b(1, 6) and (2, 3) or pairs with the opposite sign. The amount of the roots is 7 (the coefficient with a variable with the opposite sign). From here we conclude that the solutions of the square equation are x \u003d 2; x \u003d 3.
It is easier to select the roots of the equation among the free member dividers, adjusting their sign in order to fulfill the formulas of Vieta. At the beginning, it seems difficult to do, but with practice on a number of square equations, such a technique will be more effective than the calculation of the discriminant and finding the roots of the square equation to the classical method.
As you can see the school theory of study of the discriminant and the methods of finding solutions of the equation is devoid of practical meaning - "Why schoolchildren square equation?", What is the physical meaning of the discriminant? ".

Let's try to figure out what does the discriminant describe?

The course of algebra is studying the functions, research schemes of the function and the construction of the graphics of the functions. Of all the functions, a parabola occupies an important place, the equation of which can be written as
So the physical meaning of the square equation is zero parabola, that is, the intersection points of the function with the axis of the abscissa OX
Properties Parabolas that are described below will ask you to remember. The time will come to pass exams, tests, or entrance exams and you will be grateful for the reference material. The sign with a variable in the square corresponds to whether the polebola branches will be on the schedule to go up (a\u003e 0),

or parabola branches down (a<0) .

The top of Parabola lies in the middle between the roots

The physical meaning of the discriminant:

If the discriminant is greater than zero (D\u003e 0) Parabola has two intersection points with OX axis.
If the discriminant is zero (d \u003d 0), then the parabol is in the top concerns the abscissa axis.
And the last case when the discriminant is less than zero (D<0) – график параболы принадлежит плоскости над осью абсцисс (ветки параболы вверх), или график полностью под осью абсцисс (ветки параболы опущены вниз).

Incomplete square equations