The discriminant, like quadratic equations, begins to be studied in the course of algebra in the 8th grade. You can solve the quadratic equation through the discriminant and using Vieta's theorem. Study methodology quadratic equations, like the discriminant formulas, it is rather unsuccessfully inculcated in schoolchildren, like much in real education. Therefore pass school years, education in grades 9-11 replaces " higher education"and everyone is looking again - "How to solve a quadratic equation?", "How to find the roots of an equation?", "How to find the discriminant?" and...

Discriminant formula

The discriminant D of the quadratic equation a * x ^ 2 + bx + c = 0 is D = b ^ 2–4 * a * c.
The roots (solutions) of the quadratic equation depend on the sign of the discriminant (D):
D> 0 - the equation has 2 different real roots;
D = 0 - the equation has 1 root (2 coinciding roots):
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 yet, so who knows how to implement this, please write to the mail This email address is being protected from spambots. You need JavaScript enabled to view it. .

General formula for finding the roots of a quadratic equation:

We find the roots of the equation by the formula
If the coefficient of the variable squared is paired, then it is advisable to calculate not the discriminant, but its fourth part
In such cases, the roots of the equation are found by the formula

The second way to find roots is Vieta's Theorem.

A theorem is formulated not only for quadratic equations, but also for polynomials. You can read this on Wikipedia or other electronic resources. However, for simplicity, we will consider that part of it that concerns the reduced quadratic equations, that is, equations of the form (a = 1)
The essence of Vieta's formulas is that the sum of the roots of the equation is equal to the coefficient of the variable, taken with the opposite sign. The product of the roots of the equation is equal to the free term. Vieta's theorem is written in formulas.
The derivation of Vieta's formula is quite simple. Let's write the quadratic equation in terms of prime factors
As you can see, all ingenious is simple at the same time. It is effective to use the Vieta formula when the difference in the absolute values ​​of the roots or the difference in the absolute values ​​of the roots is equal to 1, 2. For example, the following equations by the Vieta theorem have roots




Up to 4 equations, the analysis should look like this. The product of the roots of the equation is 6, therefore the roots can be the values ​​(1, 6) and (2, 3) or pairs with the opposite sign. The sum of the roots is 7 (coefficient of a variable with the opposite sign). Hence we conclude that the solutions of the quadratic equation are equal to x = 2; x = 3.
It is easier to select the roots of the equation among the divisors of the free term, correcting their sign in order to fulfill the Vieta formulas. At the beginning it seems difficult to do, but with practice on a number of quadratic equations, such a technique will be more effective than calculating the discriminant and finding the roots of the quadratic equation in the classical way.
As you can see, the school theory of studying the discriminant and ways of finding solutions to the equation is devoid of practical meaning - "Why do schoolchildren need a quadratic equation?", "What is the physical meaning of the discriminant?"

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

The algebra course teaches functions, function study charts, and function graphing. Of all the functions, an important place is occupied by a parabola, the equation of which can be written in the form
So the physical meaning of the quadratic equation is the zeros of the parabola, that is, the points of intersection of the graph of the function with the abscissa axis Ox
I ask you to remember the properties of parabolas that are described below. The time will come to pass exams, tests, or entrance exams and you will be grateful for the reference material. The sign at the variable in the square corresponds to whether the branches of the parabola on the graph will go up (a> 0),

or a parabola with branches down (a<0) .

The vertex of the parabola lies in the middle between the roots

The physical meaning of the discriminant:

If the discriminant is greater than zero (D> 0), the parabola has two points of intersection with the Ox axis.
If the discriminant is equal to zero (D = 0) then the parabola at the vertex touches the abscissa axis.
And the last case when the discriminant 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 quadratic equations

Among the entire course of the school curriculum of algebra, one of the most voluminous topics is the topic of quadratic equations. In this case, a quadratic equation means an equation of the form ax 2 + bx + c = 0, where a ≠ 0 (read: and multiply by x squared plus be x plus tse is equal to zero, where a is not equal to zero). In this case, the main place is occupied by formulas for finding the discriminant of a quadratic equation of the specified type, which is understood as an expression that allows one to determine the presence or absence of roots in a quadratic equation, as well as their number (if any).

Formula (equation) of the discriminant of a quadratic equation

The generally accepted formula for the discriminant of a quadratic equation is as follows: D = b 2 - 4ac. Calculating the discriminant according to the specified formula, one can not only determine the presence and number of roots in a quadratic equation, but also choose a method for finding these roots, of which there are several depending on the type of quadratic equation.

What does it mean if the discriminant is zero \ The formula for the roots of a quadratic equation if the discriminant is zero

The discriminant, as follows from the formula, is denoted by the Latin letter D. In the case when the discriminant is zero, it should be concluded that a quadratic equation of the form ax 2 + bx + c = 0, where a ≠ 0, has only one root, which is calculated by simplified formula. This formula is applied only with zero discriminant and looks as follows: x = –b / 2a, where x is the root of the quadratic equation, b and a are the corresponding variables of the quadratic equation. To find the root of a quadratic equation, it is necessary to divide the negative value of the variable b by the doubled value of the variable a. The resulting expression will be the solution to the quadratic equation.

Solving a quadratic equation in terms of the discriminant

If, when calculating the discriminant using the above formula, a positive value is obtained (D is greater than zero), then the quadratic equation has two roots, which are calculated using the following formulas: x 1 = (–b + vD) / 2a, x 2 = (–b - vD) / 2a. Most often, the discriminant is not calculated separately, but the radical expression in the form of a discriminant formula is simply substituted into the D value from which the root is extracted. If the variable b has an even value, then to calculate the roots of a quadratic equation of the form ax 2 + bx + c = 0, where a ≠ 0, you can also use the following formulas: x 1 = (–k + v (k2 - ac)) / a , x 2 = (–k + v (k2 - ac)) / a, where k = b / 2.

In some cases, for the practical solution of quadratic equations, you can use Vieta's Theorem, which states that for the sum of the roots of a quadratic equation of the form x 2 + px + q = 0, the value x 1 + x 2 = –p will be valid, and for the product of the roots of the specified equation - expression x 1 xx 2 = q.

Can the discriminant be less than zero

When calculating the value of the discriminant, you may encounter a situation that does not fall under any of the described cases - when the discriminant has a negative value (that is, less than zero). In this case, it is customary to assume that the quadratic equation of the form ax 2 + bx + c = 0, where a ≠ 0, has no real roots, therefore, its solution will be limited to calculating the discriminant, and the above formulas for the roots of the quadratic equation in in this case will not apply. In this case, in the answer to the quadratic equation, it is written that "the equation has no real roots."

Explanatory video:

First level

Quadratic equations. Comprehensive guide (2019)

In the term "quadratic", the key word is "quadratic". This means that the equation must necessarily contain a variable (the same x) squared, and there must be no x in the third (or greater) degree.

The solution of many equations is reduced to the solution of quadratic equations.

Let's learn to determine that we have a quadratic equation, and not some other.

Example 1.

Let's get rid of the denominator and multiply each term in the equation by

Move everything to the left side and arrange the terms in descending order of the degrees of x

Now we can confidently say that this equation is quadratic!

Example 2.

Let's multiply the left and right sides by:

This equation, although it was originally in it, is not square!

Example 3.

Let's multiply everything by:

Fearfully? Fourth and second degrees ... However, if we make a substitution, then we will see that we have a simple quadratic equation:

Example 4.

It seems to be there, but let's take a closer look. Let's move everything to the left side:

You see, it has shrunk - and now it's a simple linear equation!

Now try to figure out for yourself which of the following equations are quadratic and which are not:

Examples:

Answers:

1. square;
2. square;
3. not square;
4. not square;
5. not square;
6. square;
7. not square;
8. square.

Mathematicians conditionally divide all quadratic equations into the following form:

• Complete quadratic equations- equations in which the coefficients and, as well as the free term c are not equal to zero (as in the example). In addition, among the complete quadratic equations, there are given- these are equations in which the coefficient (the equation from example one is not only complete, but also reduced!)

• Incomplete quadratic equations- equations in which the coefficient and or the free term c are equal to zero:

  They are incomplete, because they lack some element. But the equation must always have an x ​​squared !!! Otherwise, it will no longer be a square, but some other equation.

Why did you come up with such a division? It would seem that there is an X squared, and okay. This division is due to the methods of solution. Let's consider each of them in more detail.

Solving incomplete quadratic equations

First, let's dwell on solving incomplete quadratic equations - they are much simpler!

Incomplete quadratic equations are of the following types:

1. , in this equation the coefficient is.
2. , in this equation the free term is.
3. , in this equation the coefficient and the intercept are equal.

1.and. Since we know how to take the square root, let's express from this equation

The expression can be either negative or positive. The number squared cannot be negative, because when multiplying two negative or two positive numbers, the result will always be a positive number, so: if, then the equation has no solutions.

And if, then we get two roots. These formulas do not need to be memorized. The main thing is that you must know and always remember that there cannot be less.

Let's try to solve a few examples.

Example 5:

Solve the equation

Now it remains to extract the root from the left and right sides. Do you remember how to extract roots?

Answer:

Never forget about negative roots !!!

Example 6:

Solve the equation

Answer:

Example 7:

Solve the equation

Ouch! The square of a number cannot be negative, which means that the equation

no roots!

For equations that have no roots, mathematicians have come up with a special icon - (empty set). And the answer can be written like this:

Answer:

Thus, this quadratic equation has two roots. There are no restrictions here, since we did not extract the root.
Example 8:

Solve the equation

Let's take the common factor out of the parentheses:

Thus,

This equation has two roots.

Answer:

The simplest type of incomplete quadratic equations (although they are all simple, right?). Obviously, this equation always has only one root:

We'll do without examples here.

Solving complete quadratic equations

We remind you that a complete quadratic equation is an equation of the form equation where

Solving complete quadratic equations is a little more difficult (just a little) than the ones given.

Remember, any quadratic equation can be solved using the discriminant! Even incomplete.

The rest of the methods will help you do it faster, but if you have problems with quadratic equations, first learn the solution using the discriminant.

1. Solving quadratic equations using the discriminant.

Solving quadratic equations in this way is very simple, the main thing is to remember the sequence of actions and a couple of formulas.

If, then the equation has a root. You need to pay special attention to the step. The discriminant () indicates to us the number of roots of the equation.

• If, then the formula in step will be reduced to. Thus, the equation will have the entire root.
• If, then we will not be able to extract the root from the discriminant at the step. This indicates that the equation has no roots.

Let's go back to our equations and look at some examples.

Example 9:

Solve the equation

Step 1 skip.

Step 2.

We find the discriminant:

So the equation has two roots.

Step 3.

Answer:

Example 10:

Solve the equation

The equation is presented in the standard form, therefore Step 1 skip.

Step 2.

We find the discriminant:

So the equation has one root.

Answer:

Example 11:

Solve the equation

The equation is presented in the standard form, therefore Step 1 skip.

Step 2.

We find the discriminant:

Therefore, we will not be able to extract the root from the discriminant. There are no roots of the equation.

Now we know how to write down such responses correctly.

Answer: No roots

2. Solving quadratic equations using Vieta's theorem.

If you remember, there is a type of equations that are called reduced (when the coefficient a is equal):

Such equations are very easy to solve using Vieta's theorem:

Sum of roots given the quadratic equation is, and the product of the roots is.

Example 12:

Solve the equation

This equation is suitable for solving using Vieta's theorem, since ...

The sum of the roots of the equation is equal, i.e. we get the first equation:

And the product is equal to:

Let's compose and solve the system:

• and. The amount is equal;
• and. The amount is equal;
• and. The amount is equal.

and are the solution of the system:

Answer: ; .

Example 13:

Solve the equation

Answer:

Example 14:

Solve the equation

The equation is reduced, which means:

Answer:

QUADRATIC EQUATIONS. AVERAGE LEVEL

What is a Quadratic Equation?

In other words, a quadratic equation is an equation of the form, where is the unknown, are some numbers, and.

The number is called the eldest or first odds quadratic equation, - second coefficient, a - free member.

Why? Because if, the equation will immediately become linear, because disappear.

Moreover, and can be equal to zero. In this chair, the equation is called incomplete. If all the terms are in place, that is, the equation is complete.

Solutions to various types of quadratic equations

Methods for solving incomplete quadratic equations:

To begin with, let's analyze the methods for solving incomplete quadratic equations - they are simpler.

The following types of equations can be distinguished:

I., in this equation the coefficient and the intercept are equal.

II. , in this equation the coefficient is.

III. , in this equation the free term is.

Now let's look at a solution to each of these subtypes.

Obviously, this equation always has only one root:

A squared number cannot be negative, because when you multiply two negative or two positive numbers, the result will always be a positive number. That's why:

if, then the equation has no solutions;

if, we have two roots

These formulas do not need to be memorized. The main thing to remember is that it cannot be less.

Examples:

Solutions:

Answer:

Never forget negative roots!

The square of a number cannot be negative, which means that the equation

no roots.

To briefly record that the problem has no solutions, we use the empty set icon.

Answer:

So, this equation has two roots: and.

Answer:

Pull the common factor out of the parentheses:

The product is equal to zero if at least one of the factors is equal to zero. This means that the equation has a solution when:

So, this quadratic equation has two roots: and.

Example:

Solve the equation.

Solution:

Factor the left side of the equation and find the roots:

Answer:

Methods for solving complete quadratic equations:

1. Discriminant

Solving quadratic equations in this way is easy, the main thing is to remember the sequence of actions and a couple of formulas. Remember, any quadratic equation can be solved using the discriminant! Even incomplete.

Have you noticed the root of the discriminant in the root formula? But the discriminant can be negative. What to do? It is necessary to pay special attention to step 2. The discriminant indicates to us the number of roots of the equation.

• If, then the equation has a root:
• If, then the equation has the same root, but in fact, one root:

  Such roots are called double roots.

• If, then the root of the discriminant is not extracted. This indicates that the equation has no roots.

Why is there a different number of roots? Let's turn to the geometric meaning of the quadratic equation. The function graph is a parabola:

In the special case, which is a quadratic equation,. And this means that the roots of the quadratic equation are the points of intersection with the abscissa axis (axis). The parabola may not intersect the axis at all, or it may intersect it at one (when the vertex of the parabola lies on the axis) or two points.

In addition, the coefficient is responsible for the direction of the branches of the parabola. If, then the branches of the parabola are directed upward, and if - then downward.

Examples:

Solutions:

Answer:

Answer: .

Answer:

So there are no solutions.

Answer: .

2. Vieta's theorem

It is very easy to use Vieta's theorem: you just need to choose a pair of numbers, the product of which is equal to the free term of the equation, and the sum is the second coefficient, taken with the opposite sign.

It is important to remember that Vieta's theorem can only be applied in reduced quadratic equations ().

Let's look at a few examples:

Example # 1:

Solve the equation.

Solution:

This equation is suitable for solving using Vieta's theorem, since ... Other coefficients:; ...

The sum of the roots of the equation is:

And the product is equal to:

Let's select such pairs of numbers, the product of which is equal, and check whether their sum is equal:

• and. The amount is equal;
• and. The amount is equal;
• and. The amount is equal.

and are the solution of the system:

Thus, and are the roots of our equation.

Answer: ; ...

Example # 2:

Solution:

Let us select such pairs of numbers that give in the product, and then check whether their sum is equal:

and: add up.

and: add up. To get, you just need to change the signs of the alleged roots: and, after all, the product.

Answer:

Example # 3:

Solution:

The free term of the equation is negative, which means that the product of the roots is a negative number. This is only possible if one of the roots is negative and the other is positive. Therefore, the sum of the roots is difference of their modules.

Let us select such pairs of numbers that give in the product, and the difference of which is equal to:

and: their difference is equal - does not fit;

and: - does not fit;

and: - does not fit;

and: - fits. It only remains to remember that one of the roots is negative. Since their sum must be equal, then the root of the smallest in absolute value must be negative:. We check:

Answer:

Example # 4:

Solve the equation.

Solution:

The equation is reduced, which means:

The free term is negative, which means that the product of the roots is negative. And this is possible only when one root of the equation is negative and the other is positive.

Let's select such pairs of numbers, the product of which is equal, and then determine which roots should have a negative sign:

Obviously, only the roots and are suitable for the first condition:

Answer:

Example # 5:

Solve the equation.

Solution:

The equation is reduced, which means:

The sum of the roots is negative, which means that at least one of the roots is negative. But since their product is positive, then both roots are with a minus sign.

Let's select such pairs of numbers, the product of which is equal to:

Obviously, the roots are the numbers and.

Answer:

Admit it, it's very convenient to come up with roots orally, instead of counting this nasty discriminant. Try to use Vieta's theorem as often as possible.

But Vieta's theorem is needed in order to facilitate and speed up the finding of roots. To make it profitable for you to use it, you must bring the actions to automatism. And for this, decide on five more examples. But don't cheat: you can't use the discriminant! Vieta's theorem only:

Solutions for tasks for independent work:

Task 1. ((x) ^ (2)) - 8x + 12 = 0

By Vieta's theorem:

As usual, we start the selection with a piece:

Not suitable, since the amount;

: the amount is what you need.

Answer: ; ...

Task 2.

And again, our favorite Vieta theorem: the sum should work out, but the product is equal.

But since it should be not, but, we change the signs of the roots: and (in the sum).

Answer: ; ...

Task 3.

Hmm ... Where is that?

It is necessary to transfer all the terms into one part:

The sum of the roots is equal to, the product.

So stop! The equation is not given. But Vieta's theorem is applicable only in the above equations. So first you need to bring the equation. If you can't bring it up, drop this venture and solve it in another way (for example, through the discriminant). Let me remind you that to bring a quadratic equation means to make the leading coefficient equal to:

Fine. Then the sum of the roots is equal, and the product.

It's easy to pick up here: after all - a prime number (sorry for the tautology).

Answer: ; ...

Task 4.

The free term is negative. What's so special about it? And the fact that the roots will be of different signs. And now, during the selection, we check not the sum of the roots, but the difference of their modules: this difference is equal, but the product.

So, the roots are equal and, but one of them is with a minus. Vieta's theorem tells us that the sum of the roots is equal to the second coefficient with the opposite sign, that is. This means that the smaller root will have a minus: and, since.

Answer: ; ...

Task 5.

What's the first thing to do? That's right, give the equation:

Again: we select the factors of the number, and their difference should be equal to:

The roots are equal and, but one of them is with a minus. Which? Their sum must be equal, which means that with a minus there will be a larger root.

Answer: ; ...

To summarize:

1. Vieta's theorem is used only in the given quadratic equations.
2. Using Vieta's theorem, you can find the roots by selection, orally.
3. If the equation is not given or there is not a single suitable pair of free term multipliers, then there are no whole roots, and you need to solve in another way (for example, through the discriminant).

3. Method of selection of a complete square

If all the terms containing the unknown are represented in the form of terms from the abbreviated multiplication formulas - the square of the sum or difference - then, after changing the variables, the equation can be represented as an incomplete quadratic equation of the type.

For example:

Example 1:

Solve the equation:.

Solution:

Answer:

Example 2:

Solve the equation:.

Solution:

Answer:

In general, the transformation will look like this:

This implies: .

Doesn't it look like anything? This is a discriminant! That's right, we got the discriminant formula.

QUADRATIC EQUATIONS. BRIEFLY ABOUT THE MAIN

Quadratic equation is an equation of the form, where is the unknown, are the coefficients of the quadratic equation, is the free term.

Full quadratic equation- an equation in which the coefficients are not equal to zero.

Reduced quadratic equation- an equation in which the coefficient, that is:.

Incomplete Quadratic Equation- an equation in which the coefficient and or the free term c are equal to zero:

• if the coefficient, the equation has the form:,
• if the free term, the equation has the form:,
• if and, the equation has the form:.

1. Algorithm for solving incomplete quadratic equations

1.1. Incomplete quadratic equation of the form, where,:

1) Let us express the unknown:,

2) Check the sign of the expression:

• if, then the equation has no solutions,
• if, then the equation has two roots.

1.2. Incomplete quadratic equation of the form, where,:

1) Pull the common factor out of the brackets:,

2) The product is equal to zero if at least one of the factors is equal to zero. Therefore, the equation has two roots:

1.3. Incomplete quadratic equation of the form, where:

This equation always has only one root:.

2. Algorithm for solving complete quadratic equations of the form where

2.1. Decision using the discriminant

1) Let us bring the equation to the standard form:,

2) We calculate the discriminant by the formula:, which indicates the number of roots of the equation:

3) Find the roots of the equation:

• if, then the equation has roots, which are found by the formula:
• if, then the equation has a root, which is found by the formula:
• if, then the equation has no roots.

2.2. Solution using Vieta's theorem

The sum of the roots of the reduced quadratic equation (equations of the form, where) is equal, and the product of the roots is equal, i.e. , a.

2.3. Full square solution

For example, for the trinomial \ (3x ^ 2 + 2x-7 \), the discriminant will be \ (2 ^ 2-4 \ cdot3 \ cdot (-7) = 4 + 84 = 88 \). And for the trinomial \ (x ^ 2-5x + 11 \), it will be \ ((- 5) ^ 2-4 \ cdot1 \ cdot11 = 25-44 = -19 \).

The discriminant is denoted by the letter \ (D \) and is often used when solving. Also, by the value of the discriminant, you can understand how the graph looks approximately (see below).

Discriminant and roots of the equation

The discriminant value shows the amount of the quadratic equation:
- if \ (D \) is positive - the equation will have two roots;
- if \ (D \) is equal to zero - only one root;
- if \ (D \) is negative, there are no roots.

This does not need to be learned, it is easy to come to this conclusion, just knowing what from the discriminant (that is, \ (\ sqrt (D) \) enters the formula for calculating the roots of the equation: \ (x_ (1) = \) \ (\ frac (-b + \ sqrt (D)) (2a) \) and \ (x_ (2) = \) \ (\ frac (-b- \ sqrt (D)) (2a) \) Let's take a closer look at each case ...

If the discriminant is positive

In this case, the root of it is some positive number, which means \ (x_ (1) \) and \ (x_ (2) \) will be different in meaning, because in the first formula \ (\ sqrt (D) \) is added , and in the second, it is subtracted. And we have two different roots.

Example : Find the roots of the equation \ (x ^ 2 + 2x-3 = 0 \)
Solution :

Answer : \ (x_ (1) = 1 \); \ (x_ (2) = - 3 \)

If the discriminant is zero

And how many roots will there be if the discriminant is zero? Let's reason.

The root formulas look like this: \ (x_ (1) = \) \ (\ frac (-b + \ sqrt (D)) (2a) \) and \ (x_ (2) = \) \ (\ frac (-b- \ sqrt (D)) (2a) \). And if the discriminant is zero, then the root of it is also zero. Then it turns out:

\ (x_ (1) = \) \ (\ frac (-b + \ sqrt (D)) (2a) \) \ (= \) \ (\ frac (-b + \ sqrt (0)) (2a) \) \ (= \) \ (\ frac (-b + 0) (2a) \) \ (= \) \ (\ frac (-b) (2a) \)

\ (x_ (2) = \) \ (\ frac (-b- \ sqrt (D)) (2a) \) \ (= \) \ (\ frac (-b- \ sqrt (0)) (2a) \) \ (= \) \ (\ frac (-b-0) (2a) \) \ (= \) \ (\ frac (-b) (2a) \)

That is, the values ​​of the roots of the equation will be the same, because adding or subtracting zero does not change anything.

Example : Find the roots of the equation \ (x ^ 2-4x + 4 = 0 \)
Solution :

\ (x ^ 2-4x + 4 = 0 \)		We write out the coefficients:
\ (a = 1; \) \ (b = -4; \) \ (c = 4; \)		Calculate the discriminant by the formula \ (D = b ^ 2-4ac \)
\ (D = (- 4) ^ 2-4 \ cdot1 \ cdot4 = \) \(=16-16=0\)		Find the roots of the equation
\ (x_ (1) = \) \ (\ frac (- (- 4) + \ sqrt (0)) (2 \ cdot1) \)\ (= \) \ (\ frac (4) (2) \) \ (= 2 \) \ (x_ (2) = \) \ (\ frac (- (- 4) - \ sqrt (0)) (2 \ cdot1) \)\ (= \) \ (\ frac (4) (2) \) \ (= 2 \)		We got two identical roots, so it makes no sense to write them separately - we write them down as one.

Answer : \ (x = 2 \)

Quadratic equation - easy to solve! * Further in the text "KU". Friends, it would seem, what could be easier in mathematics than solving such an equation. But something told me that many have problems with him. I decided to see how many impressions per month Yandex. Here's what happened, take a look:

What does it mean? This means that about 70,000 people a month are looking for this information, and what will happen in the middle of the academic year - there will be twice as many requests. This is not surprising, because those guys and girls who graduated from school a long time ago and are preparing for the Unified State Exam are looking for this information, and schoolchildren also seek to refresh it in their memory.

Despite the fact that there are a lot of sites that tell you how to solve this equation, I decided to do my bit too and publish the material. Firstly, I want visitors to come to my site for this request; secondly, in other articles, when the "KU" speech comes, I will give a link to this article; thirdly, I will tell you about his solution a little more than is usually stated on other sites. Let's get started! The content of the article:

A quadratic equation is an equation of the form:

where the coefficients a,band with arbitrary numbers, with a ≠ 0.

V school course the material is given in the following form - the equations are conditionally divided into three classes:

1. They have two roots.

2. * Have only one root.

3. Have no roots. It is worth noting here that they have no valid roots.

How are roots calculated? Just!

We calculate the discriminant. Underneath this "terrible" word lies a very simple formula:

The root formulas are as follows:

* These formulas need to be known by heart.

You can immediately write down and decide:

Example:

1. If D> 0, then the equation has two roots.

2. If D = 0, then the equation has one root.

3. If D< 0, то уравнение не имеет действительных корней.

Let's take a look at the equation:

In this regard, when the discriminant is zero, in the school course it is said that one root is obtained, here it is equal to nine. Everything is correct, it is, but ...

This representation is somewhat incorrect. In fact, there are two roots. Yes, yes, do not be surprised, it turns out two equal roots, and to be mathematically exact, then the answer should be written two roots:

x 1 = 3 x 2 = 3

But this is so - a small digression. At school, you can write down and say that there is one root.

Now the next example:

As we know, the root of negative number is not retrieved, so there is no solution in this case.

That's the whole solution process.

Quadratic function.

Here's how the solution looks geometrically. This is extremely important to understand (in the future, in one of the articles, we will analyze in detail the solution of the square inequality).

This is a function of the form:

where x and y are variables

a, b, c - given numbers, with a ≠ 0

The graph is a parabola:

That is, it turns out that by solving the quadratic equation with "y" equal to zero, we find the points of intersection of the parabola with the x-axis. There can be two of these points (the discriminant is positive), one (the discriminant is zero) and none (the discriminant is negative). Details about quadratic function You can view article by Inna Feldman.

Let's consider some examples:

Example 1: Solve 2x 2 +8 x–192=0

a = 2 b = 8 c = –192

D = b 2 –4ac = 8 2 –4 ∙ 2 ∙ (–192) = 64 + 1536 = 1600

Answer: x 1 = 8 x 2 = –12

* It was possible to immediately divide the left and right sides of the equation by 2, that is, to simplify it. The calculations will be easier.

Example 2: Decide x 2–22 x + 121 = 0

a = 1 b = –22 c = 121

D = b 2 –4ac = (- 22) 2 –4 ∙ 1 ∙ 121 = 484–484 = 0

We got that x 1 = 11 and x 2 = 11

In the answer, it is permissible to write x = 11.

Answer: x = 11

Example 3: Decide x 2 –8x + 72 = 0

a = 1 b = –8 c = 72

D = b 2 –4ac = (- 8) 2 –4 ∙ 1 ∙ 72 = 64–288 = –224

The discriminant is negative, there is no solution in real numbers.

Answer: no solution

The discriminant is negative. There is a solution!

Here we will talk about solving the equation in the case when a negative discriminant is obtained. Do you know anything about complex numbers? I will not go into detail here about why and where they came from and what their specific role and need in mathematics are, this is a topic for a large separate article.

The concept of a complex number.

A bit of theory.

A complex number z is a number of the form

z = a + bi

where a and b are real numbers, i is the so-called imaginary unit.

a + bi Is a SINGLE NUMBER, not addition.

The imaginary unit is equal to the root of minus one:

Now consider the equation:

We got two conjugate roots.

Incomplete quadratic equation.

Consider special cases, this is when the coefficient "b" or "c" is equal to zero (or both are equal to zero). They are easily solved without any discriminants.

Case 1. Coefficient b = 0.

The equation takes the form:

Let's transform:

Example:

4x 2 –16 = 0 => 4x 2 = 16 => x 2 = 4 => x 1 = 2 x 2 = –2

Case 2. Coefficient with = 0.

The equation takes the form:

We transform, factorize:

* The product is equal to zero when at least one of the factors is equal to zero.

Example:

9x 2 –45x = 0 => 9x (x – 5) = 0 => x = 0 or x – 5 = 0

x 1 = 0 x 2 = 5

Case 3. Coefficients b = 0 and c = 0.

It is clear here that the solution to the equation will always be x = 0.

Useful properties and patterns of coefficients.

There are properties that allow you to solve equations with large coefficients.

ax 2 + bx+ c=0 equality holds

a + b+ c = 0, then

- if for the coefficients of the equation ax 2 + bx+ c=0 equality holds

a+ c =b, then

These properties help to solve a certain kind of equation.

Example 1: 5001 x 2 –4995 x – 6=0

The sum of the odds is 5001+ ( – 4995)+(– 6) = 0, hence

Example 2: 2501 x 2 +2507 x+6=0

Equality is met a+ c =b, means

Regularities of the coefficients.

1. If in the equation ax 2 + bx + c = 0 the coefficient "b" is equal to (a 2 +1), and the coefficient "c" is numerically equal to the coefficient "a", then its roots are

ax 2 + (a 2 +1) ∙ х + а = 0 => х 1 = –а х 2 = –1 / a.

Example. Consider the equation 6x 2 + 37x + 6 = 0.

x 1 = –6 x 2 = –1/6.

2. If in the equation ax 2 - bx + c = 0 the coefficient "b" is equal to (a 2 +1), and the coefficient "c" is numerically equal to the coefficient "a", then its roots are

ax 2 - (a 2 +1) ∙ x + a = 0 => x 1 = a x 2 = 1 / a.

Example. Consider the equation 15x 2 –226x +15 = 0.

x 1 = 15 x 2 = 1/15.

3. If in the equation ax 2 + bx - c = 0 coefficient "b" is equal to (a 2 - 1), and the coefficient "c" numerically equal to the coefficient "a", then its roots are equal

ax 2 + (a 2 –1) ∙ х - а = 0 => х 1 = - а х 2 = 1 / a.

Example. Consider the equation 17x 2 + 288x - 17 = 0.

x 1 = - 17 x 2 = 1/17.

4. If in the equation ax 2 - bx - c = 0 the coefficient "b" is equal to (a 2 - 1), and the coefficient c is numerically equal to the coefficient "a", then its roots are

аx 2 - (а 2 –1) ∙ х - а = 0 => х 1 = а х 2 = - 1 / a.

Example. Consider the equation 10x 2 - 99x –10 = 0.

x 1 = 10 x 2 = - 1/10

Vieta's theorem.

Vieta's theorem is named after the famous French mathematician Francois Vieta. Using Vieta's theorem, one can express the sum and product of the roots of an arbitrary KE in terms of its coefficients.

45 = 1∙45 45 = 3∙15 45 = 5∙9.

In total, the number 14 gives only 5 and 9. These are the roots. With a certain skill, using the presented theorem, you can solve many quadratic equations verbally.

Vieta's theorem, moreover. convenient in that after solving the quadratic equation in the usual way (through the discriminant), the obtained roots can be checked. I recommend doing this at all times.

TRANSFER METHOD

With this method, the coefficient "a" is multiplied by the free term, as if "thrown" to it, therefore it is called by means of "transfer". This method is used when you can easily find the roots of the equation using Vieta's theorem and, most importantly, when the discriminant is an exact square.

If a± b + c≠ 0, then the transfer technique is used, for example:

2NS 2 – 11x + 5 = 0 (1) => NS 2 – 11x + 10 = 0 (2)

By Vieta's theorem in equation (2) it is easy to determine that x 1 = 10 x 2 = 1

The obtained roots of the equation must be divided by 2 (since two were "thrown" from x 2), we get

x 1 = 5 x 2 = 0.5.

What is the rationale? See what's going on.

The discriminants of equations (1) and (2) are equal:

If you look at the roots of the equations, then only different denominators are obtained, and the result depends precisely on the coefficient at x 2:

The second (modified) roots are 2 times larger.

Therefore, we divide the result by 2.

* If we re-roll a three, then we divide the result by 3, etc.

Answer: x 1 = 5 x 2 = 0.5

Sq. ur-ye and exam.

I will say briefly about its importance - YOU MUST BE ABLE TO SOLVE quickly and without hesitation, the formulas of the roots and the discriminant must be known by heart. A lot of the tasks that make up the USE tasks are reduced to solving a quadratic equation (including geometric ones).

What is worth noting!

1. The form of writing the equation can be "implicit". For example, the following entry is possible:

15+ 9x 2 - 45x = 0 or 15x + 42 + 9x 2 - 45x = 0 or 15 -5x + 10x 2 = 0.

You need to bring it to a standard form (so as not to get confused when solving).

2. Remember that x is an unknown quantity and it can be denoted by any other letter - t, q, p, h and others.