Quadratic formula in terms of k. Square root: calculation formulas. The formula for finding the roots of a quadratic equation. Relationship between roots and coefficients of a quadratic equation

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

General view of the quadratic equation

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

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

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

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

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

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

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

Types of records of quadratic equations

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

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

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

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

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

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

How is a complete quadratic equation solved?

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

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

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

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

How is an incomplete quadratic equation solved?

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

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

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

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

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

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

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

Examples

It is required to solve the following quadratic equations:

x 2 - 7x \u003d 0;

15 - 2x - x 2 \u003d 0;

x 2 + 8 + 3x = 0;

12x + x 2 + 36 = 0;

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

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

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

The first root takes on the value: x 1 \u003d 0. The second will be found from the linear equation: x - 7 \u003d 0. It is easy to see that x 2 \u003d 7.

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

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

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

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

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

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

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:

The solution of equations in mathematics occupies a special place. This process is preceded by many hours of studying the theory, during which the student learns how to solve equations, determine their form and bring the skill to full automatism. However, the search for roots does not always make sense, since they may simply not exist. There are special methods for finding roots. In this article, we will analyze the main functions, their domains of definition, as well as cases where 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 such number x, the product of which 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 parts are presented 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 value . Which equation has no roots? Examples of linear equations are shown in the illustration below.

Basically, linear equations are solved by simply transferring the numerical part to one part and the contents of x to the other. It turns out an equation of the form mx \u003d n, where m and n are numbers, and x is an unknown. To find x, it is enough to divide both parts by m. Then x = n/m. Basically, linear equations have only one root, but there are cases when there are either infinitely many roots or none at all. When m = 0 and n = 0, the equation takes the form 0 * x = 0. Absolutely any number will be the solution to such an equation.

But what equation has no roots?

For m = 0 and n = 0, the equation has no roots from 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 \u003d 0 for a \u003d 0. The most common is the solution through the discriminant. The formula for finding the discriminant of a quadratic equation: D \u003d 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 on the graph of a function, which is a parabola. For a > 0, the branches are directed upwards, 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 top of the parabola and determine in which direction the branches are directed. You can determine the x-coordinate of the vertex by the formula: x 0 \u003d -b / 2a. In this case, the y-coordinate of the vertex is found by simply substituting the x0 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, hence 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 a trigonometric circle with radius 1, |sinx| and |cosx| cannot be greater than 1. So which sinx equation has no roots? Consider the sinx function graph shown in the picture below.

We see that the function is symmetrical 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 it is greater than one in absolute value.

This is the simplest example of trigonometric equations. In fact, their solution can take many pages, at the end of which you realize that you used the wrong formula and you need to start all over again. Sometimes, even with the correct finding of the roots, you can forget to take into account the restrictions on the ODZ, which is why an extra root or interval appears in the answer, and the whole answer turns into an erroneous one. 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 combined with curly or square brackets. Curly braces denote the joint execution of all equations. That is, if at least one of the equations has no roots or contradicts the other, the whole system has no solution. Square brackets denote the word "or". This means that if at least one of the equations of the system has a solution, then the whole system has a solution.

The answer of 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 this complexity does not allow you to immediately say which equation has no roots.

In problem books and textbooks, there are different types of equations: those that have roots, and those that do not have them. First of all, if you can't find roots, don't think they don't exist at all. Perhaps you have made a mistake somewhere, then it is enough to carefully double-check your decision.

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

So, the equation has no roots if:

in the linear equation mx = n, the value m = 0 and n = 0;
in a quadratic equation if the discriminant is less than zero;
in a 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.

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

Content

See also: Solving quadratic equations online

Basic Formulas

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

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

.

Graphic interpretation

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

Useful Formulas Related to Quadratic Equation

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

Derivation of the formula for the roots of a quadratic equation

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

,
where
; .

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

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

Examples of determining the roots of a quadratic equation

Example 1

(1.1) .

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

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

.

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

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

;
;
.

Example 2

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

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

Then the factorization of the trinomial has the form:
.

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

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

;
.

Example 3

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

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

You can find complex roots:
;
;
.

Then

.