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 tons 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 a little more about its solution 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.

In the 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 do not have valid roots.

How are roots calculated? Just!

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

The root formulas are as follows:

* You need to know these formulas 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 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, 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. It is extremely important to understand this (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 ox 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 look at 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

аx 2 + (а 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 equal

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 François Vieta. Using Vieta's theorem, we 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 the "transfer" method. 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 tasks of the exam 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.

With this math program, you can solve quadratic equation.

The program not only gives an answer to the problem, but also displays the solution process in two ways:
- using the discriminant
- using Vieta's theorem (if possible).

Moreover, the answer is displayed exact, not approximate.
For example, for the equation \ (81x ^ 2-16x-1 = 0 \), the answer is displayed in this form:

$$ x_1 = \ frac (8+ \ sqrt (145)) (81), \ quad x_2 = \ frac (8- \ sqrt (145)) (81) $$ and not like this: \ (x_1 = 0.247; \ quad x_2 = -0.05 \)

This program can be useful for high school students in preparation for control works and exams, when checking knowledge before the exam, parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to do as quickly as possible homework in math or algebra? In this case, you can also use our programs with a detailed solution.

In this way, you can conduct your own training and / or the training of your younger brothers or sisters, while the level of education in the field of the problems being solved rises.

If you are not familiar with the rules for entering a square polynomial, we recommend that you familiarize yourself with them.

Rules for entering a square polynomial

Any Latin letter can be used as a variable.
For example: \ (x, y, z, a, b, c, o, p, q \) etc.

Numbers can be entered as whole or fractional numbers.
Moreover, fractional numbers can be entered not only in the form of a decimal, but also in the form of an ordinary fraction.

Rules for entering decimal fractions.
In decimal fractions, the fractional part from the whole can be separated by either a point or a comma.
For example, you can enter decimals so: 2.5x - 3.5x ^ 2

Rules for entering ordinary fractions.
Only an integer can be used as the numerator, denominator and whole part of a fraction.

The denominator cannot be negative.

When entering a numeric fraction, the numerator is separated from the denominator by a division sign: /
Whole part separated from the fraction by an ampersand: &
Input: 3 & 1/3 - 5 & 6 / 5z + 1 / 7z ^ 2
Result: \ (3 \ frac (1) (3) - 5 \ frac (6) (5) z + \ frac (1) (7) z ^ 2 \)

When entering an expression brackets can be used... In this case, when solving a quadratic equation, the introduced expression is first simplified.
For example: 1/2 (y-1) (y + 1) - (5y-10 & 1/2)

=0

Decide

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A bit of theory.

Quadratic equation and its roots. Incomplete quadratic equations

Each of the equations
\ (- x ^ 2 + 6x + 1,4 = 0, \ quad 8x ^ 2-7x = 0, \ quad x ^ 2- \ frac (4) (9) = 0 \)
has the form
\ (ax ^ 2 + bx + c = 0, \)
where x is a variable, a, b and c are numbers.
In the first equation a = -1, b = 6 and c = 1.4, in the second a = 8, b = -7 and c = 0, in the third a = 1, b = 0 and c = 4/9. Such equations are called quadratic equations.

Definition.
Quadratic equation is an equation of the form ax 2 + bx + c = 0, where x is a variable, a, b and c are some numbers, and \ (a \ neq 0 \).

The numbers a, b and c are the coefficients of the quadratic equation. The number a is called the first coefficient, the number b - the second coefficient, and the number c - the free term.

In each of the equations of the form ax 2 + bx + c = 0, where \ (a \ neq 0 \), the greatest power of the variable x is the square. Hence the name: quadratic equation.

Note that a quadratic equation is also called an equation of the second degree, since its left side is a polynomial of the second degree.

A quadratic equation in which the coefficient at x 2 is 1 is called reduced quadratic equation... For example, the reduced quadratic equations are the equations
\ (x ^ 2-11x + 30 = 0, \ quad x ^ 2-6x = 0, \ quad x ^ 2-8 = 0 \)

If in the quadratic equation ax 2 + bx + c = 0 at least one of the coefficients b or c is equal to zero, then such an equation is called incomplete quadratic equation... So, the equations -2x 2 + 7 = 0, 3x 2 -10x = 0, -4x 2 = 0 are incomplete quadratic equations. In the first of them b = 0, in the second c = 0, in the third b = 0 and c = 0.

Incomplete quadratic equations are of three types:
1) ax 2 + c = 0, where \ (c \ neq 0 \);
2) ax 2 + bx = 0, where \ (b \ neq 0 \);
3) ax 2 = 0.

Let's consider the solution of equations of each of these types.

To solve an incomplete quadratic equation of the form ax 2 + c = 0 for \ (c \ neq 0 \), transfer its free term to the right side and divide both sides of the equation by a:
\ (x ^ 2 = - \ frac (c) (a) \ Rightarrow x_ (1,2) = \ pm \ sqrt (- \ frac (c) (a)) \)

Since \ (c \ neq 0 \), then \ (- \ frac (c) (a) \ neq 0 \)

If \ (- \ frac (c) (a)> 0 \), then the equation has two roots.

If \ (- \ frac (c) (a) To solve an incomplete quadratic equation of the form ax 2 + bx = 0 with \ (b \ neq 0 \) factor its left side into factors and obtain the equation
\ (x (ax + b) = 0 \ Rightarrow \ left \ (\ begin (array) (l) x = 0 \\ ax + b = 0 \ end (array) \ right. \ Rightarrow \ left \ (\ begin (array) (l) x = 0 \\ x = - \ frac (b) (a) \ end (array) \ right. \)

Hence, an incomplete quadratic equation of the form ax 2 + bx = 0 for \ (b \ neq 0 \) always has two roots.

An incomplete quadratic equation of the form ax 2 = 0 is equivalent to the equation x 2 = 0 and therefore has a unique root 0.

The formula for the roots of a quadratic equation

Let us now consider how quadratic equations are solved in which both the coefficients of the unknowns and the free term are nonzero.

Solve the quadratic equation in general view and as a result we get the formula for the roots. Then this formula can be applied to solve any quadratic equation.

Solve the quadratic equation ax 2 + bx + c = 0

Dividing both of its parts by a, we obtain the equivalent reduced quadratic equation
\ (x ^ 2 + \ frac (b) (a) x + \ frac (c) (a) = 0 \)

We transform this equation by selecting the square of the binomial:
\ (x ^ 2 + 2x \ cdot \ frac (b) (2a) + \ left (\ frac (b) (2a) \ right) ^ 2- \ left (\ frac (b) (2a) \ right) ^ 2 + \ frac (c) (a) = 0 \ Rightarrow \)

\ (x ^ 2 + 2x \ cdot \ frac (b) (2a) + \ left (\ frac (b) (2a) \ right) ^ 2 = \ left (\ frac (b) (2a) \ right) ^ 2 - \ frac (c) (a) \ Rightarrow \) \ (\ left (x + \ frac (b) (2a) \ right) ^ 2 = \ frac (b ^ 2) (4a ^ 2) - \ frac ( c) (a) \ Rightarrow \ left (x + \ frac (b) (2a) \ right) ^ 2 = \ frac (b ^ 2-4ac) (4a ^ 2) \ Rightarrow \) \ (x + \ frac (b ) (2a) = \ pm \ sqrt (\ frac (b ^ 2-4ac) (4a ^ 2)) \ Rightarrow x = - \ frac (b) (2a) + \ frac (\ pm \ sqrt (b ^ 2 -4ac)) (2a) \ Rightarrow \) \ (x = \ frac (-b \ pm \ sqrt (b ^ 2-4ac)) (2a) \)

The radical expression is called the discriminant of the quadratic equation ax 2 + bx + c = 0 (Latin "discriminant" is a discriminator). It is designated by the letter D, i.e.
\ (D = b ^ 2-4ac \)

Now, using the notation of the discriminant, we rewrite the formula for the roots of the quadratic equation:
\ (x_ (1,2) = \ frac (-b \ pm \ sqrt (D)) (2a) \), where \ (D = b ^ 2-4ac \)

It's obvious that:
1) If D> 0, then the quadratic equation has two roots.
2) If D = 0, then the quadratic equation has one root \ (x = - \ frac (b) (2a) \).
3) If D Thus, depending on the value of the discriminant, the quadratic equation can have two roots (for D> 0), one root (for D = 0) or not have roots (for D When solving a quadratic equation using this formula, it is advisable to proceed as follows way:
1) calculate the discriminant and compare it with zero;
2) if the discriminant is positive or equal to zero, then use the root formula, if the discriminant is negative, then write down that there are no roots.

Vieta's theorem

The given quadratic equation ax 2 -7x + 10 = 0 has roots 2 and 5. The sum of the roots is 7, and the product is 10. We see that the sum of the roots is equal to the second coefficient taken with the opposite sign, and the product of the roots is equal to the free term. Any given quadratic equation that has roots has this property.

The sum of the roots of the given quadratic equation is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term.

Those. Vieta's theorem states that the roots x 1 and x 2 of the reduced quadratic equation x 2 + px + q = 0 have the property:
\ (\ left \ (\ begin (array) (l) x_1 + x_2 = -p \\ x_1 \ cdot x_2 = q \ end (array) \ right. \)

Quadratic equations are studied in grade 8, so there is nothing complicated here. The ability to solve them is absolutely essential.

A quadratic equation is an equation of the form ax 2 + bx + c = 0, where the coefficients a, b and c are arbitrary numbers, and a ≠ 0.

Before studying specific methods for solving, we note that all quadratic equations can be conditionally divided into three classes:

1. Have no roots;
2. Have exactly one root;
3. They have two distinct roots.

This is an important difference. quadratic equations from linear, where the root always exists and is unique. How do you determine how many roots an equation has? There is a wonderful thing for this - discriminant.

Discriminant

Let a quadratic equation ax 2 + bx + c = 0 be given. Then the discriminant is just the number D = b 2 - 4ac.

You need to know this formula by heart. Where it comes from - it doesn't matter now. Another thing is important: by the sign of the discriminant, you can determine how many roots a quadratic equation has. Namely:

1. If D< 0, корней нет;
2. If D = 0, there is exactly one root;
3. If D> 0, there will be two roots.

Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many believe. Take a look at the examples - and you yourself will understand everything:

Task. How many roots do quadratic equations have:

1. x 2 - 8x + 12 = 0;
2. 5x 2 + 3x + 7 = 0;
3. x 2 - 6x + 9 = 0.

Let us write down the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 - 4 1 12 = 64 - 48 = 16

So the discriminant is positive, so the equation has two different roots. We analyze the second equation in a similar way:
a = 5; b = 3; c = 7;
D = 3 2 - 4 5 7 = 9 - 140 = −131.

The discriminant is negative, there are no roots. The last equation remains:
a = 1; b = −6; c = 9;
D = (−6) 2 - 4 1 9 = 36 - 36 = 0.

The discriminant is zero - there will be one root.

Note that coefficients have been written for each equation. Yes, it’s long, yes, it’s boring - but you won’t mix up the coefficients and don’t make stupid mistakes. Choose for yourself: speed or quality.

By the way, if you “fill your hand”, after a while you will no longer need to write out all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 equations are solved - in general, not that much.

Quadratic Roots

Now let's move on to the solution. If the discriminant D> 0, the roots can be found by the formulas:

Basic formula for the roots of a quadratic equation

When D = 0, you can use any of these formulas - you get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.

1. x 2 - 2x - 3 = 0;
2. 15 - 2x - x 2 = 0;
3. x 2 + 12x + 36 = 0.

First equation:
x 2 - 2x - 3 = 0 ⇒ a = 1; b = −2; c = −3;
D = (−2) 2 - 4 1 (−3) = 16.

D> 0 ⇒ the equation has two roots. Let's find them:

Second equation:
15 - 2x - x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 - 4 (−1) 15 = 64.

D> 0 ⇒ the equation has two roots again. Let's find them

\ [\ begin (align) & ((x) _ (1)) = \ frac (2+ \ sqrt (64)) (2 \ cdot \ left (-1 \ right)) = - 5; \\ & ((x) _ (2)) = \ frac (2- \ sqrt (64)) (2 \ cdot \ left (-1 \ right)) = 3. \\ \ end (align) \]

Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 - 4 · 1 · 36 = 0.

D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:

As you can see from the examples, everything is very simple. If you know the formulas and be able to count, there will be no problems. Most often, errors occur when substituting negative coefficients in the formula. Here, again, the technique described above will help: look at the formula literally, describe each step - and very soon you will get rid of mistakes.

Incomplete quadratic equations

It happens that the quadratic equation is somewhat different from what is given in the definition. For example:

1. x 2 + 9x = 0;
2. x 2 - 16 = 0.

It is easy to see that one of the terms is missing in these equations. Such quadratic equations are even easier to solve than standard ones: they do not even need to calculate the discriminant. So, let's introduce a new concept:

The equation ax 2 + bx + c = 0 is called an incomplete quadratic equation if b = 0 or c = 0, i.e. coefficient at variable x or free element is equal to zero.

Of course, a very difficult case is possible when both of these coefficients are equal to zero: b = c = 0. In this case, the equation takes the form ax 2 = 0. Obviously, such an equation has a single root: x = 0.

Let's consider the rest of the cases. Let b = 0, then we get an incomplete quadratic equation of the form ax 2 + c = 0. Let's transform it a little:

Since the arithmetic square root exists only from a non-negative number, the last equality makes sense only for (−c / a) ≥ 0. Conclusion:

1. If the inequality (−c / a) ≥ 0 holds in an incomplete quadratic equation of the form ax 2 + c = 0, there will be two roots. The formula is given above;
2. If (−c / a)< 0, корней нет.

As you can see, the discriminant was not required - in incomplete quadratic equations there are no complicated calculations at all. In fact, it is not even necessary to remember the inequality (−c / a) ≥ 0. It is enough to express the value x 2 and see what stands on the other side of the equal sign. If there is a positive number, there will be two roots. If negative, there will be no roots at all.

Now let's deal with equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factor out the polynomial:

Bracketing a common factor

The product is zero when at least one of the factors is zero. From here are the roots. In conclusion, we will analyze several such equations:

Task. Solve quadratic equations:

1. x 2 - 7x = 0;
2. 5x 2 + 30 = 0;
3. 4x 2 - 9 = 0.

x 2 - 7x = 0 ⇒ x (x - 7) = 0 ⇒ x 1 = 0; x 2 = - (- 7) / 1 = 7.

5x 2 + 30 = 0 ⇒ 5x 2 = −30 ⇒ x 2 = −6. There are no roots, tk. a square cannot be equal to a negative number.

4x 2 - 9 = 0 ⇒ 4x 2 = 9 ⇒ x 2 = 9/4 ⇒ x 1 = 3/2 = 1.5; x 2 = −1.5.

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

General view of the quadratic equation

Here, their explicit recording is proposed, when the highest degree is recorded first, and then in descending order. There are often situations when the terms are out of order. Then it is better to rewrite the equation in decreasing order of the degree of the variable.

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

If we accept these designations, all quadratic equations are reduced to the following record.

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 there will be in the answer. Because one of three options is always possible:

• there will be two roots in the solution;
• the answer is one number;
• the equation will have no roots at all.

And until 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 contain their different records. They will not always look like general formula quadratic equation. Sometimes it will lack some terms. What was written above is a 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 in 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. Formulas for an incomplete form of equations will be as follows:

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

Discriminant and dependence of the number of roots on its value

You need to know this number in order to calculate the roots of the equation. It can always be calculated, no matter what the formula for the quadratic equation. 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 has been found 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 this formula.

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

Formula number five. The same record shows 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. There is even no need for additional formulas. And you will not need those that have already been recorded for the discriminant and the unknown.

First, consider the incomplete equation number two. In this equality, it is supposed to take the unknown quantity out of the parenthesis and solve the linear equation, which remains in the parentheses. 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.

Incomplete equation 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 factor in front of the unknown. All that remains is to extract the square root and remember to write it down twice with opposite signs.

Next, some actions are written to help you learn how to solve all kinds of equalities that turn into quadratic equations. They will help the student to avoid careless mistakes. These flaws are the reason bad grades when studying the extensive topic "Quadratic equations (grade 8)". Subsequently, these actions will not need to be constantly performed. Because a stable skill will appear.

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

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

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

Examples of

It is required to solve the following quadratic equations:

x 2 - 7x = 0;

15 - 2x - x 2 = 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 = 0. It is incomplete, therefore it is solved as described for the formula number two.

After leaving the brackets, it turns out: x (x - 7) = 0.

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

Second equation: 5x 2 + 30 = 0. Again incomplete. Only it is solved as described for the third formula.

After transferring 30 to the right side of the equality: 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.

The third equation: 15 - 2x - x 2 = 0. Hereinafter, the solution of quadratic equations will begin with their rewriting into standard view: - x 2 - 2x + 15 = 0. Now it's time to use the second useful advice and multiply everything by minus one. It turns out x 2 + 2x - 15 = 0. According to the fourth formula, you need to calculate the discriminant: D = 2 2 - 4 * (- 15) = 4 + 60 = 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 using the fifth formula. It turns out that x = (-2 ± √64) / 2 = (-2 ± 8) / 2. Then x 1 = 3, x 2 = - 5.

The fourth equation x 2 + 8 + 3x = 0 is transformed into this: x 2 + 3x + 8 = 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 = -12 / (2 * 1) = -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 similar terms, before opening the brackets. In place of the first, there will be such an expression: x 2 + 2x + 1. After the equality, this record will appear: x 2 + 3x + 2. After such terms are counted, the equation will take the form: x 2 - x = 0. It turned into incomplete ... Something similar to it has already been considered a little higher. The roots of this will be the numbers 0 and 1.

We continue to study the topic “ solving equations". We have already met with linear equations and are moving on to get acquainted with quadratic equations.

First, we will analyze 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, let's move on to the solution complete equations, we obtain the formula for the roots, get acquainted with the discriminant of the quadratic equation and consider the solutions of typical examples. Finally, let's trace the relationship 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 related definitions. 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 nonzero.

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.

The 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 the highest, or the coefficient at x 2, b is the second coefficient, or the coefficient at x, and c is the free term.

For example, let's take a quadratic equation of the form 5x2 −2x3 = 0, here the leading coefficient is 5, the second coefficient is −2, and the intercept is −3. Note that when the coefficients b and / or c are negative, as in the example just given, then we use short form writing a quadratic equation of the form 5 x 2 −2 x − 3 = 0, and 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 quadratic equation, which is due to the peculiarities of writing such. For example, in a quadratic equation y 2 −y + 3 = 0, the leading coefficient is one, and the coefficient at y is −1.

Reduced and unreduced quadratic equations

Reduced and non-reduced quadratic equations are distinguished depending on the value of the leading coefficient. 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, quadratic equations x 2 −3 x + 1 = 0, x 2 −x − 2/3 = 0, etc. - given, 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 parts of it 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 unreduced quadratic equation, or, like it, has no roots.

Let us analyze by example 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 divide both sides of the original equation by the leading factor 3, it is nonzero, so we can perform this action. We have (3 x 2 + 12 x − 7): 3 = 0: 3, which is the same, (3 x 2): 3+ (12 x): 3−7: 3 = 0, and beyond (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

The definition of a quadratic equation contains the condition a ≠ 0. This condition is necessary in order for the equation a x 2 + b x + c = 0 to be exactly quadratic, since at 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 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.

Full quadratic equation Is an equation in which all coefficients are nonzero.

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

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 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

From the information in the previous paragraph it follows that there is three kinds of incomplete quadratic equations:

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

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

a x 2 = 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 both parts of it by a nonzero number a. Obviously, the root of the equation x 2 = 0 is zero, since 0 2 = 0. This equation has no other roots, which is explained, indeed, for any nonzero number p, the inequality p 2> 0 holds, whence it follows that for p ≠ 0 the equality p 2 = 0 is never achieved.

So, the incomplete quadratic equation a · x 2 = 0 has a single root x = 0.

As an example, let us give the solution to the incomplete quadratic equation −4 · x 2 = 0. Equation x 2 = 0 is equivalent to it, its only root is x = 0, therefore, the original equation also has a unique root zero.

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

a x 2 + c = 0

Now let us consider how incomplete quadratic equations are solved, in which the coefficient b is zero, and c ≠ 0, that is, equations of the form a · x 2 + c = 0. We know that transferring a term from one side of the equation to another with the opposite sign, as well as dividing both sides of the equation by a nonzero number, give an equivalent equation. Therefore, it is possible to carry out the following equivalent transformations of the incomplete quadratic equation a x 2 + c = 0:

• move c to the right-hand side, which gives the equation a x 2 = −c,
• and divide both of 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 , since by hypothesis c ≠ 0. Let us examine separately 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 you remember about, then the root of the equation immediately becomes obvious, it is a 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 the contradictory method. Let's do it.

Let us denote the roots of the equation just sounded as x 1 and −x 1. Suppose that the equation has one more root x 2, different from the indicated roots x 1 and −x 1. It is known that substitution of its roots into an equation instead of x 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 actions with numbers allow you to rewrite the resulting equality as (x 1 - x 2) · (x 1 + x 2) = 0. We know that the product of two numbers is zero if and only if at least one of them is 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. This is how we came 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 roots other than and.

Let's summarize the information of this item. The incomplete quadratic equation a x 2 + c = 0 is equivalent to the equation that

• 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.

Solve another incomplete quadratic equation −x 2 + 9 = 0. Move the nine to the right: −x 2 = −9. Now we divide both sides by −1, we get x 2 = 9. On the right side there is a positive number, from which we conclude that or. Then 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 factor out the common factor x. This allows us to pass 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 combination 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.

Moving x out of parentheses gives the equation. It is equivalent to two equations x = 0 and. We solve the resulting linear equation:, and performing division mixed number on common fraction, we find. Therefore, the roots of the original equation are x = 0 and.

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

Answer:

x = 0,.

Discriminant, the formula for the roots of a quadratic equation

There is a root formula for solving quadratic equations. Let's write down quadratic formula: , where D = b 2 −4 a c- so-called quadratic discriminant... The notation essentially means that.

It is useful to know how the root formula was obtained, and how it is applied when finding the roots of quadratic equations. Let's figure it out.

Derivation of the formula for the roots of a quadratic equation

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

• We can divide both sides of this equation by a nonzero number a, as a result we get the reduced quadratic equation.
• Now select a complete 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-hand side with the opposite sign, we have.
• And we also transform the expression on the right side:.

As a result, we come to an equation that 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 them. 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, whence its only root is visible;
• if, then or, which is the same 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 was called the discriminant of the 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.

Returning to the equation, rewrite it using the discriminant notation:. And we draw conclusions:

• 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 by virtue can be rewritten in the form or, and after expanding and reducing the fractions to a common denominator, we obtain.

So we derived formulas for the roots of a quadratic equation, they have the form, 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 the 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 of a negative number, which takes us out of the box and school curriculum... With a negative discriminant, the quadratic equation has no real roots, but has a pair complex conjugate roots, which can be found by the same root formulas obtained by us.

Algorithm for solving quadratic equations using root formulas

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

However, in a school algebra course, usually it comes not about complex, but about real roots of the 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 only after that calculate the values ​​of the roots.

The above reasoning allows us to write quadratic equation solver... To solve the quadratic equation a x 2 + b x + c = 0, you need:

• by 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 by the formula if D = 0;
• find two real roots of a quadratic equation using the root formula if the discriminant is positive.

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

You can proceed to examples of using the algorithm for solving quadratic equations.

Examples of solving quadratic equations

Consider solutions to three quadratic equations with positive, negative and zero discriminants. 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, first you 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, then the quadratic equation has two real roots. We find them using the root formula, we get, here you can simplify the expressions obtained by doing factoring out the sign of the root with the subsequent reduction of the fraction:

Answer:

Let's move on to the next typical example.

Example.

Solve the quadratic equation −4x2 + 28x − 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 it is necessary to indicate complex roots, then we apply the well-known formula for the roots of the quadratic equation, and perform complex number operations:

Answer:

there are no real roots, complex roots are as follows:.

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

Root formula for even second coefficients

The formula for the roots of a quadratic equation, where D = b 2 −4 ln5 = 2 7 ln5). Let's take it 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, 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 formula for roots:

Let us denote the expression n 2 −a · c as D 1 (sometimes it is denoted by 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 of D 1 is also an indicator of the presence or absence of the roots of a quadratic equation.

So, to solve the 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 by the formula;
• If D 1> 0, then find two real roots by the formula.

Consider solving an example using the root formula obtained in this paragraph.

Example.

Solve the quadratic equation 5x2 −6x − 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. Let's 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:

Simplifying the View of Quadratic Equations

Sometimes, before embarking on the calculation of the roots of a quadratic equation by 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 parts of it by some number. For example, in the previous paragraph, we managed to simplify the equation 1100x2 −400x − 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 sides 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. the absolute values ​​of its coefficients: GCD (12, 42, 48) = GCD (GCD (12, 42), 48) = GCD (6, 48) = 6. Dividing both sides 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 sides of the quadratic equation is usually done to get rid of fractional coefficients. In this case, the multiplication is carried out by the denominators of its coefficients. For example, if both sides of the quadratic equation are multiplied by the LCM (6, 3, 1) = 6, then it will take the 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 leading coefficient of the quadratic equation, changing the signs of all terms, which corresponds to multiplying (or dividing) both parts by −1. For example, usually from the quadratic equation −2x2 −3x + 7 = 0 one goes over to the solution 2x2 + 3x − 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 for the roots, you can get other relationships between the roots and the coefficients.

The best known and most applicable formulas are from Vieta's 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 equal to the free term. For example, by the form of the quadratic equation 3 x 2 −7 x + 22 = 0, we 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 the coefficients of the quadratic equation. For example, you can express the sum of the squares of the roots of a quadratic equation through its coefficients:.

Bibliography.

• Algebra: study. for 8 cl. 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.
• A. G. Mordkovich Algebra. 8th grade. At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., Erased. - M .: Mnemosina, 2009 .-- 215 p .: ill. ISBN 978-5-346-01155-2.