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

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 solutions, note that all quadratic equations can be divided into three classes:

1. Have no roots;
2. Have exactly one root;
3. Have two various roots.

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

Discriminant

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

You need to know this formula by heart. Where it comes from is not important 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 people believe. Take a look at the examples and you will understand everything yourself:

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's write out 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 left is:
a = 1; b = −6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.

Discriminant equal to zero- there will be one root.

Please note that coefficients have been written down for each equation. Yes, it’s long, yes, it’s tedious, but you won’t mix up the odds and make stupid mistakes. Choose for yourself: speed or quality.

By the way, if you get the hang of it, after a while you won’t need to write down all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not that much.

Roots of a quadratic equation

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

Basic root formula quadratic equation

When D = 0, you can use any of these formulas - you will 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 again has two roots. 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 can count, there will be no problems. Most often, errors occur when substituting negative coefficients into the formula. Here again, the technique described above will help: look at the formula literally, write down each step - and very soon you will get rid of mistakes.

Incomplete quadratic equations

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

1. x 2 + 9x = 0;
2. x 2 − 16 = 0.

It is easy to notice that these equations are missing one of the terms. Such quadratic equations are even easier to solve than standard ones: they don’t even require calculating 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. the coefficient of the variable x or the 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 remaining cases. Let b = 0, then we obtain an incomplete quadratic equation of the form ax 2 + c = 0. Let us transform it a little:

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

1. If in an incomplete quadratic equation of the form ax 2 + c = 0 the inequality (−c /a) ≥ 0 is satisfied, 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 is no complex calculations. 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 is on the other side of the equal sign. If there positive number- there will be two roots. If it is negative, there will be no roots at all.

Now let's look at 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 the polynomial:

Removal common multiplier out of bracket

The product is zero when at least one of the factors is zero. This is where the roots come from. In conclusion, let’s look at a few of these 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, because 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.

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

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

Discriminant and roots of the equation

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

This does not need to be taught, it is not difficult to come to such a conclusion, simply knowing that from the discriminant (that is, \(\sqrt(D)\) is included in 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 look at each case in more detail .

If the discriminant is positive

In this case, the root of it is some positive number, which means \(x_(1)\) and \(x_(2)\) will have different meanings, 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

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 its root 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:
\(a=1;\) \(b=-4;\) \(c=4;\)		We calculate the discriminant using the formula \(D=b^2-4ac\)
\(D=(-4)^2-4\cdot1\cdot4=\) \(=16-16=0\)		Finding the roots of the equation
\(x_(1)=\) \(\frac(-(-4)+\sqrt(0))(2\cdot1)\)\(=\)\(\frac(4)(2)\) \(=2\) \(x_(2)=\) \(\frac(-(-4)-\sqrt(0))(2\cdot1)\)\(=\)\(\frac(4)(2)\) \(=2\)		Got two identical roots, so there is no point in writing them separately - we write them as one.

Answer : \(x=2\)

Quadratic equations. Discriminant. Solution, examples.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

Types of quadratic equations

What is a quadratic equation? What does it look like? In term quadratic equation the keyword is "square". This means that in the equation Necessarily there must be an x ​​squared. In addition to it, the equation may (or may not!) contain just X (to the first power) and just a number (free member). And there should be no X's to a power greater than two.

In mathematical terms, a quadratic equation is an equation of the form:

Here a, b and c- some numbers. b and c- absolutely any, but A– anything other than zero. For example:

Here A =1; b = 3; c = -4

Here A =2; b = -0,5; c = 2,2

Here A =-3; b = 6; c = -18

Well, you understand...

In these quadratic equations on the left there is full set members. X squared with a coefficient A, x to the first power with coefficient b And free member s.

Such quadratic equations are called full.

And if b= 0, what do we get? We have X will be lost to the first power. This happens when multiplied by zero.) It turns out, for example:

5x 2 -25 = 0,

2x 2 -6x=0,

-x 2 +4x=0

And so on. And if both coefficients b And c are equal to zero, then it’s even simpler:

2x 2 =0,

-0.3x 2 =0

Such equations where something is missing are called incomplete quadratic equations. Which is quite logical.) Please note that x squared is present in all equations.

By the way, why A can't be equal to zero? And you substitute instead A zero.) Our X squared will disappear! The equation will become linear. And the solution is completely different...

That's all the main types of quadratic equations. Complete and incomplete.

Solving quadratic equations.

Solving complete quadratic equations.

Quadratic equations are easy to solve. According to formulas and clear, simple rules. At the first stage it is necessary given equation lead to standard view, i.e. to the form:

If the equation is already given to you in this form, you do not need to do the first stage.) The main thing is to correctly determine all the coefficients, A, b And c.

The formula for finding the roots of a quadratic equation looks like this:

The expression under the root sign is called discriminant. But more about him below. As you can see, to find X, we use only a, b and c. Those. coefficients from a quadratic equation. Just carefully substitute the values a, b and c We calculate into this formula. Let's substitute with your own signs! For example, in the equation:

A =1; b = 3; c= -4. Here we write it down:

The example is almost solved:

This is the answer.

Everything is very simple. And what, you think it’s impossible to make a mistake? Well, yes, how...

The most common mistakes are confusion with sign values a, b and c. Or rather, not with their signs (where to get confused?), but with substitution negative values into the formula for calculating the roots. What helps here is a detailed recording of the formula with specific numbers. If there are problems with calculations, do that!

Suppose we need to solve the following example:

Here a = -6; b = -5; c = -1

Let's say you know that you rarely get answers the first time.

Well, don't be lazy. It will take about 30 seconds to write an extra line. And the number of errors will decrease sharply. So we write in detail, with all the brackets and signs:

It seems incredibly difficult to write out so carefully. But it only seems so. Give it a try. Well, or choose. What's better, fast or right? Besides, I will make you happy. After a while, there will be no need to write everything down so carefully. It will work out right on its own. Especially if you use practical techniques that are described below. This evil example with a bunch of minuses can be solved easily and without errors!

But, often, quadratic equations look slightly different. For example, like this:

Did you recognize it?) Yes! This incomplete quadratic equations.

Solving incomplete quadratic equations.

They can also be solved using a general formula. You just need to understand correctly what they are equal to here. a, b and c.

Have you figured it out? In the first example a = 1; b = -4; A c? It's not there at all! Well yes, that's right. In mathematics this means that c = 0 ! That's all. Substitute zero into the formula instead c, and we will succeed. Same with the second example. Only we don’t have zero here With, A b !

But incomplete quadratic equations can be solved much more simply. Without any formulas. Let's consider the first incomplete equation. What can you do on the left side? You can take X out of brackets! Let's take it out.

And what from this? And the fact that the product equals zero if and only if any of the factors equals zero! Don't believe me? Okay, then come up with two non-zero numbers that, when multiplied, will give zero!
Does not work? That's it...
Therefore, we can confidently write: x 1 = 0, x 2 = 4.

All. These will be the roots of our equation. Both are suitable. When substituting any of them into the original equation, we get the correct identity 0 = 0. As you can see, the solution is much simpler than using the general formula. Let me note, by the way, which X will be the first and which will be the second - absolutely indifferent. It is convenient to write in order, x 1- what is smaller and x 2- that which is greater.

The second equation can also be solved simply. Move 9 to the right side. We get:

All that remains is to extract the root from 9, and that’s it. It will turn out:

Also two roots . x 1 = -3, x 2 = 3.

This is how all incomplete quadratic equations are solved. Either by placing X out of brackets, or by simply moving the number to the right and then extracting the root.
It is extremely difficult to confuse these techniques. Simply because in the first case you will have to extract the root of X, which is somehow incomprehensible, and in the second case there is nothing to take out of brackets...

Discriminant. Discriminant formula.

Magic word discriminant ! Rarely a high school student has not heard this word! The phrase “we solve through a discriminant” inspires confidence and reassurance. Because there is no need to expect tricks from the discriminant! It is simple and trouble-free to use.) I remind you of the most general formula for solutions any quadratic equations:

The expression under the root sign is called a discriminant. Typically the discriminant is denoted by the letter D. Discriminant formula:

D = b 2 - 4ac

And what is so remarkable about this expression? Why did it deserve a special name? What the meaning of the discriminant? After all -b, or 2a in this formula they don’t specifically call it anything... Letters and letters.

Here's the thing. When solving a quadratic equation using this formula, it is possible only three cases.

1. The discriminant is positive. This means the root can be extracted from it. Whether the root is extracted well or poorly is another question. What is important is what is extracted in principle. Then your quadratic equation has two roots. Two various solutions.

2. The discriminant is zero. Then you will have one solution. Since adding or subtracting zero in the numerator does not change anything. Strictly speaking, this is not one root, but two identical. But, in simplified version, it is customary to talk about one solution.

3. The discriminant is negative. The square root of a negative number cannot be taken. Well, okay. This means there are no solutions.

Honestly speaking, when simple solution quadratic equations, the concept of a discriminant is not particularly required. We substitute the values ​​of the coefficients into the formula and count. Everything happens there by itself, two roots, one, and none. However, when solving more difficult tasks, without knowledge meaning and formula of the discriminant not enough. Especially in equations with parameters. Such equations are aerobatics for the State Examination and the Unified State Exam!)

So, how to solve quadratic equations through the discriminant you remembered. Or you learned, which is also not bad.) You know how to correctly determine a, b and c. Do you know how? attentively substitute them into the root formula and attentively count the result. Did you understand that keyword Here - attentively?

Now take note of practical techniques that dramatically reduce the number of errors. The same ones that are due to inattention... For which it later becomes painful and offensive...

First appointment . Don’t be lazy before solving a quadratic equation and bring it to standard form. What does this mean?
Let's say that after all the transformations you get the following equation:

Don't rush to write the root formula! You'll almost certainly get the odds mixed up a, b and c. Construct the example correctly. First, X squared, then without square, then the free term. Like this:

And again, don’t rush! A minus in front of an X squared can really upset you. It's easy to forget... Get rid of the minus. How? Yes, as taught in the previous topic! We need to multiply the entire equation by -1. We get:

But now you can safely write down the formula for the roots, calculate the discriminant and finish solving the example. Decide for yourself. You should now have roots 2 and -1.

Reception second. Check the roots! According to Vieta's theorem. Don't be scared, I'll explain everything! Checking last thing the equation. Those. the one we used to write down the root formula. If (as in this example) the coefficient a = 1, checking the roots is easy. It is enough to multiply them. The result should be a free member, i.e. in our case -2. Please note, not 2, but -2! Free member with your sign . If it doesn’t work out, it means they’ve already screwed up somewhere. Look for the error.

If it works, you need to add the roots. Last and final check. The coefficient should be b With opposite familiar. In our case -1+2 = +1. A coefficient b, which is before the X, is equal to -1. So, everything is correct!
It’s a pity that this is so simple only for examples where x squared is pure, with a coefficient a = 1. But at least check in such equations! All less mistakes will.

Reception third . If your equation has fractional odds, - get rid of fractions! Multiply the equation by common denominator, as described in the lesson "How to solve equations? Identical transformations." When working with fractions, errors keep creeping in for some reason...

By the way, I promised to simplify the evil example with a bunch of minuses. Please! Here he is.

In order not to get confused by the minuses, we multiply the equation by -1. We get:

That's all! Solving is a pleasure!

So, let's summarize the topic.

Practical advice:

1. Before solving, we bring the quadratic equation to standard form and build it Right.

2. If there is a negative coefficient in front of the X squared, we eliminate it by multiplying the entire equation by -1.

3. If the coefficients are fractional, we eliminate the fractions by multiplying the entire equation by the corresponding factor.

4. If x squared is pure, its coefficient equal to one, the solution can be easily verified using Vieta's theorem. Do it!

Now we can decide.)

Solve equations:

8x 2 - 6x + 1 = 0

x 2 + 3x + 8 = 0

x 2 - 4x + 4 = 0

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

Answers (in disarray):

x 1 = 0
x 2 = 5

x 1.2 =2

x 1 = 2
x 2 = -0.5

x - any number

x 1 = -3
x 2 = 3

no solutions

x 1 = 0.25
x 2 = 0.5

Does everything fit? Great! Quadratic equations are not your thing headache. The first three worked, but the rest didn’t? Then the problem is not with quadratic equations. The problem is in identical transformations of equations. Take a look at the link, it's helpful.

Doesn't quite work out? Or does it not work out at all? Then Section 555 will help you. All these examples are broken down there. Shown main errors in the solution. Of course, it also talks about the use identity transformations in solving various equations. Helps a lot!

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

I hope, having studied this article, you will learn to find the roots of a complete quadratic equation.

Using the discriminant, only complete quadratic equations are solved; to solve incomplete quadratic equations, other methods are used, which you will find in the article “Solving incomplete quadratic equations.”

What quadratic equations are called complete? This equations of the form ax 2 + b x + c = 0, where coefficients a, b and c are not equal to zero. So, to solve a complete quadratic equation, we need to calculate the discriminant D.

D = b 2 – 4ac.

Depending on the value of the discriminant, we will write down the answer.

If the discriminant a negative number(D< 0),то корней нет.

If the discriminant is zero, then x = (-b)/2a. When the discriminant is a positive number (D > 0),

then x 1 = (-b - √D)/2a, and x 2 = (-b + √D)/2a.

For example. Solve the equation x 2– 4x + 4= 0.

D = 4 2 – 4 4 = 0

x = (- (-4))/2 = 2

Answer: 2.

Solve Equation 2 x 2 + x + 3 = 0.

D = 1 2 – 4 2 3 = – 23

Answer: no roots.

Solve Equation 2 x 2 + 5x – 7 = 0.

D = 5 2 – 4 2 (–7) = 81

x 1 = (-5 - √81)/(2 2)= (-5 - 9)/4= – 3.5

x 2 = (-5 + √81)/(2 2) = (-5 + 9)/4=1

Answer: – 3.5; 1.

So let’s imagine the solution of complete quadratic equations using the diagram in Figure 1.

Using these formulas you can solve any complete quadratic equation. You just need to be careful to the equation was written as a polynomial of the standard form

A x 2 + bx + c, otherwise you may make a mistake. For example, in writing the equation x + 3 + 2x 2 = 0, you can mistakenly decide that

a = 1, b = 3 and c = 2. Then

D = 3 2 – 4 1 2 = 1 and then the equation has two roots. And this is not true. (See solution to example 2 above).

Therefore, if the equation is not written as a polynomial of the standard form, first the complete quadratic equation must be written as a polynomial of the standard form (the monomial with the largest exponent should come first, that is A x 2 , then with less – bx and then a free member With.

When solving the reduced quadratic equation and a quadratic equation with an even coefficient in the second term, you can use other formulas. Let's get acquainted with these formulas. If in a complete quadratic equation the second term has an even coefficient (b = 2k), then you can solve the equation using the formulas shown in the diagram in Figure 2.

A complete quadratic equation is called reduced if the coefficient at x 2 is equal to one and the equation takes the form x 2 + px + q = 0. Such an equation can be given for solution, or it can be obtained by dividing all coefficients of the equation by the coefficient A, standing at x 2 .

Figure 3 shows a diagram for solving the reduced square
equations. Let's look at an example of the application of the formulas discussed in this article.

Example. Solve the equation

3x 2 + 6x – 6 = 0.

Let's solve this equation using the formulas shown in the diagram in Figure 1.

D = 6 2 – 4 3 (– 6) = 36 + 72 = 108

√D = √108 = √(36 3) = 6√3

x 1 = (-6 - 6√3)/(2 3) = (6 (-1- √(3)))/6 = –1 – √3

x 2 = (-6 + 6√3)/(2 3) = (6 (-1+ √(3)))/6 = –1 + √3

Answer: –1 – √3; –1 + √3

You can notice that the coefficient of x in this equation even number, that is, b = 6 or b = 2k, whence k = 3. Then let’s try to solve the equation using the formulas given in the diagram of the figure D 1 = 3 2 – 3 · (– 6) = 9 + 18 = 27

√(D 1) = √27 = √(9 3) = 3√3

x 1 = (-3 - 3√3)/3 = (3 (-1 - √(3)))/3 = – 1 – √3

x 2 = (-3 + 3√3)/3 = (3 (-1 + √(3)))/3 = – 1 + √3

Answer: –1 – √3; –1 + √3. Noticing that all the coefficients in this quadratic equation are divisible by 3 and performing the division, we get the reduced quadratic equation x 2 + 2x – 2 = 0 Solve this equation using the formulas for the reduced quadratic
equations figure 3.

D 2 = 2 2 – 4 (– 2) = 4 + 8 = 12

√(D 2) = √12 = √(4 3) = 2√3

x 1 = (-2 - 2√3)/2 = (2 (-1 - √(3)))/2 = – 1 – √3

x 2 = (-2 + 2√3)/2 = (2 (-1+ √(3)))/2 = – 1 + √3

Answer: –1 – √3; –1 + √3.

As we see, when solving this equation by various formulas we received the same answer. Therefore, having thoroughly mastered the formulas shown in the diagram in Figure 1, you will always be able to solve any complete quadratic equation.

website, when copying material in full or in part, a link to the source is required.

Quadratic equations often appear during solution various tasks physics and mathematics. In this article we will look at how to solve these equalities in a universal way"through the discriminant". Examples of using the acquired knowledge are also given in the article.

What equations will we be talking about?

The figure below shows a formula in which x is an unknown variable and latin characters a, b, c represent some known numbers.

Each of these symbols is called a coefficient. As you can see, the number "a" appears before the variable x squared. This maximum degree given expression, so it is called a quadratic equation. Its other name is often used: second-order equation. The value of a itself is square coefficient(standing at the variable squared), b is linear coefficient(it is located next to the variable raised to the first power), finally, the number c is the free term.

Note that the form of the equation shown in the figure above is the general classical quadratic expression. In addition to it, there are other second-order equations in which the coefficients b and c can be zero.

When the task is set to solve the equality in question, this means that such values ​​of the variable x need to be found that would satisfy it. The first thing you need to remember here is next thing: since the maximum power of X is 2, then this type expressions cannot have more than 2 solutions. This means that if, when solving an equation, 2 values ​​of x were found that satisfy it, then you can be sure that there is no 3rd number, substituting it for x, the equality would also be true. The solutions to an equation in mathematics are called its roots.

Methods for solving second order equations

Solving equations of this type requires knowledge of some theory about them. IN school course algebras consider 4 various methods solutions. Let's list them:

• using factorization;
• using the formula for a perfect square;
• by applying the graph of the corresponding quadratic function;
• using the discriminant equation.

The advantage of the first method is its simplicity; however, it cannot be used for all equations. The second method is universal, but somewhat cumbersome. The third method is distinguished by its clarity, but it is not always convenient and applicable. And finally, using the discriminant equation is a universal and fairly simple way to find the roots of absolutely any second-order equation. Therefore, in this article we will consider only it.

Formula for obtaining the roots of the equation

Let's turn to general appearance quadratic equation. Let's write it down: a*x²+ b*x + c =0. Before using the method of solving it “through a discriminant,” you should always bring the equality to its written form. That is, it must consist of three terms (or less if b or c is 0).

For example, if there is an expression: x²-9*x+8 = -5*x+7*x², then you should first move all its terms to one side of the equality and add the terms containing the variable x in the same powers.

IN in this case this operation will lead to the following expression: -6*x²-4*x+8=0, which is equivalent to the equation 6*x²+4*x-8=0 (here we multiplied the left and right sides of the equality by -1).

In the example above, a = 6, b=4, c=-8. Note that all terms of the equality under consideration are always summed together, so if the “-” sign appears, this means that the corresponding coefficient is negative, like the number c in this case.

Having examined this point, let us now move on to the formula itself, which makes it possible to obtain the roots of a quadratic equation. It looks like the one shown in the photo below.

As can be seen from this expression, it allows you to get two roots (pay attention to the “±” sign). To do this, it is enough to substitute the coefficients b, c, and a into it.

The concept of a discriminant

IN previous paragraph a formula was given that allows you to quickly solve any second-order equation. In it, the radical expression is called a discriminant, that is, D = b²-4*a*c.

Why is this part of the formula highlighted, and it even has proper name? The fact is that the discriminant connects all three coefficients of the equation into a single expression. Last fact means that it completely carries information about the roots, which can be expressed in the following list:

1. D>0: The equality has 2 different solutions, both of which are real numbers.
2. D=0: The equation has only one root, and it is a real number.

Discriminant determination task

Let's give a simple example of how to find a discriminant. Let the following equality be given: 2*x² - 4+5*x-9*x² = 3*x-5*x²+7.

Let's bring it to standard form, we get: (2*x²-9*x²+5*x²) + (5*x-3*x) + (- 4-7) = 0, from which we come to the equality: -2*x² +2*x-11 = 0. Here a=-2, b=2, c=-11.

Now you can use the above formula for the discriminant: D = 2² - 4*(-2)*(-11) = -84. The resulting number is the answer to the task. Since in the example the discriminant is less than zero, we can say that this quadratic equation does not have real roots. Its solution will be only numbers of complex type.

An example of inequality through a discriminant

Let's solve problems of a slightly different type: given the equality -3*x²-6*x+c = 0. It is necessary to find values ​​of c for which D>0.

In this case, only 2 out of 3 coefficients are known, so it is not possible to calculate the exact value of the discriminant, but it is known that it is positive. We use the last fact when composing the inequality: D= (-6)²-4*(-3)*c>0 => 36+12*c>0. Solving the resulting inequality leads to the result: c>-3.

Let's check the resulting number. To do this, we calculate D for 2 cases: c=-2 and c=-4. The number -2 satisfies the obtained result (-2>-3), the corresponding discriminant will have the value: D = 12>0. In turn, the number -4 does not satisfy the inequality (-4. Thus, any numbers c that are greater than -3 will satisfy the condition.

An example of solving an equation

Let us present a problem that involves not only finding the discriminant, but also solving the equation. It is necessary to find the roots for the equality -2*x²+7-9*x = 0.

In this example, the discriminant is equal to the following value: D = 81-4*(-2)*7= 137. Then the roots of the equation are determined as follows: x = (9±√137)/(-4). This exact values roots, if you calculate the root approximately, then you get the numbers: x = -5.176 and x = 0.676.

Geometric problem

We will solve a problem that will require not only the ability to calculate the discriminant, but also the application of skills abstract thinking and knowledge of how to write quadratic equations.

Bob had a 5 x 4 meter duvet. The boy wanted to sew it around the entire perimeter continuous strip from beautiful fabric. How thick will this strip be if we know that Bob has 10 m² of fabric.

Let the strip have a thickness of x m, then the area of ​​the fabric along the long side of the blanket will be (5+2*x)*x, and since there are 2 long sides, we have: 2*x*(5+2*x). On the short side, the area of ​​​​the sewn fabric will be 4*x, since there are 2 of these sides, we get the value 8*x. Note that the value 2*x was added to the long side because the length of the blanket increased by that number. The total area of ​​fabric sewn to the blanket is 10 m². Therefore, we get the equality: 2*x*(5+2*x) + 8*x = 10 => 4*x²+18*x-10 = 0.

For this example, the discriminant is equal to: D = 18²-4*4*(-10) = 484. Its root is 22. Using the formula, we find the required roots: x = (-18±22)/(2*4) = (- 5; 0.5). Obviously, of the two roots, only the number 0.5 is suitable according to the conditions of the problem.

Thus, the strip of fabric that Bob sews to his blanket will be 50 cm wide.