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 are "very even ...")

Types of quadratic equations

What is a Quadratic Equation? What does it look like? In term quadratic equation the key word is "square". It means that in the equation necessarily there must be an x ​​squared. In addition to him, the equation may (or may not be!) Just x (in the first power) and just a number (free member). And there should be no x's to a degree greater than two.

Mathematically speaking, 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 get the idea ...

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

Such quadratic equations are called full.

What if b= 0, what do we get? We have X will disappear in the first degree. This happens from multiplication by zero.) It turns out, for example:

5x 2 -25 = 0,

2x 2 -6x = 0,

-x 2 + 4x = 0

Etc. And if both coefficients, b and c are equal to zero, it is still 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 the x squared is present in all equations.

By the way, why a can't be zero? And you substitute a zero.) The X in the square will disappear from us! The equation becomes linear. And it is decided in a completely different way ...

These are 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 to bring the given equation to a standard form, i.e. to look:

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:

An expression under the root sign is called discriminant... But about him - below. As you can see, to find x, we use only a, b and c. Those. coefficients from the quadratic equation. Just carefully substitute the values a, b and c into this formula and count. Substitute with your signs! For example, in the equation:

a =1; b = 3; c= -4. So we write down:

The example is almost solved:

This is the answer.

Everything is very simple. And what, you think, is impossible to be mistaken? Well, yes, how ...

The most common mistakes are confusion with meaning signs. a, b and c... Rather, not with their signs (where to get confused there?), But with the substitution of negative values ​​in the formula for calculating the roots. Here, a detailed notation of the formula with specific numbers saves. If there are computational problems, do so!

Suppose you need to solve this 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 30 seconds to write an extra line. And the number of errors will sharply decrease... So we write in detail, with all the brackets and signs:

It seems incredibly difficult to paint so carefully. But it only seems to be. Try it. Well, or choose. Which is better, fast, or right? Besides, I will make you happy. After a while, there will be no need to paint everything so carefully. It will work out right by itself. Especially if you use the practical techniques described below. This evil example with a bunch of drawbacks can be solved easily and without errors!

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

Did you find out?) Yes! it incomplete quadratic equations.

Solving incomplete quadratic equations.

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

Have you figured it out? In the first example a = 1; b = -4; a c? He's not there at all! Well, yes, that's right. In mathematics, this means that c = 0 ! That's all. Substitute zero in the formula instead of c, and we will succeed. The same is with the second example. Only zero we have here not with, a b !

But incomplete quadratic equations can be solved much easier. Without any formulas. Consider the first incomplete equation. What can you do there on the left side? You can put the x out of the brackets! Let's take it out.

And what of it? And the fact that the product is equal to zero if, and only if, when any of the factors is equal to zero! Don't believe me? Well, then think of 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.

Everything. These will be the roots of our equation. Both fit. When substituting any of them into the original equation, we get the correct identity 0 = 0. As you can see, the solution is much easier than using the general formula. I will note, by the way, which X will be the first, and which will be the second - it is absolutely indifferent. It is convenient to write down in order, x 1- what is less, and x 2- what is more.

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

It remains 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 the x in parentheses, 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 from the x, which is somehow incomprehensible, and in the second case there is nothing to put out of brackets ...

Discriminant. Discriminant formula.

Magic word discriminant ! A rare high school student has not heard this word! The phrase “deciding through the discriminant” is reassuring and reassuring. Because there is no need to wait for dirty tricks from the discriminant! It is simple and trouble-free to use.) I recall the most general formula for solving any quadratic equations:

The expression under the root sign is called the discriminant. Usually 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 do not specifically name ... 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 you can extract the root from it. Good root is extracted, or bad - another question. It is important what is extracted in principle. Then your quadratic equation has two roots. Two different solutions.

2. The discriminant is zero. Then you have one solution. Since the addition-subtraction of zero in the numerator does not change anything. Strictly speaking, this is not one root, but two identical... But, in a simplified version, it is customary to talk about one solution.

3. The discriminant is negative. No square root is extracted from a negative number. Well, okay. This means that there are no solutions.

Honestly, with a simple solution of quadratic equations, the concept of the discriminant is not particularly required. We substitute the values ​​of the coefficients into the formula, but we count. There, everything turns out by itself, and two roots, and one, and not one. However, when solving more complex tasks, without knowledge meaning and discriminant formulas not enough. Especially - in equations with parameters. Such equations are aerobatics at the State Examination and the Unified State Exam!)

So, how to solve quadratic equations through the discriminant you remembered. Or have learned, which is also not bad.) You know how to correctly identify a, b and c... You know how attentively substitute them in the root formula and attentively read the result. You get the idea that the key word here is attentively?

For now, take note of the best practices that will drastically reduce errors. The very ones that are due to inattention. ... For which then it hurts and insults ...

First reception ... Do not be lazy to bring it to the standard form before solving the quadratic equation. What does this mean?
Let's say, after some transformations, you got the following equation:

Don't rush to write the root formula! You will almost certainly mix up the odds. a, b and c. Build the example correctly. First, the X is squared, then without the square, then the free member. Like this:

And again, do not rush! The minus in front of the x in the square can make you really sad. It's easy to forget it ... Get rid of the minus. How? Yes, as taught in the previous topic! You have to multiply the whole equation by -1. We get:

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

Reception of the second. Check the roots! By Vieta's theorem. Do not be alarmed, I will explain everything! Checking last thing the equation. Those. the one by which we wrote down the formula for the roots. If (as in this example) the coefficient a = 1, checking the roots is easy. It is enough to multiply them. You should get a free member, i.e. in our case, -2. Pay attention, not 2, but -2! Free member with my sign ... If it didn’t work, then it’s already screwed up somewhere. Look for a bug.

If it works out, you need to fold the roots. The last and final check. You should get a coefficient b with opposite familiar. In our case, -1 + 2 = +1. And the coefficient b which is before the x is -1. So, everything is correct!
It is a pity that this is so simple only for examples where the x squared is pure, with a coefficient a = 1. But at least in such equations, check! There will be fewer mistakes.

Reception third ... If your equation has fractional coefficients, get rid of fractions! Multiply the equation by the common denominator as described in the lesson How to Solve Equations? Identical Transformations. When working with fractions, for some reason, errors come in ...

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

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

That's all! It's a pleasure to decide!

So, to summarize the topic.

Practical advice:

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

2. If there is a negative coefficient in front of the x in the square, 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 appropriate factor.

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

Now you 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 it all fit together? Fine! Quadratic equations are not your 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 walk on the link, it's helpful.

Not quite working out? Or does it not work at all? Then Section 555 will help you. There all these examples are sorted out to pieces. Shown the main errors in the solution. Of course, it also talks about the use of identical transformations in the solution of 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. Instant validation testing. Learning - with interest!)

you can get acquainted with functions and derivatives.

5x (x - 4) = 0

5 x = 0 or x - 4 = 0

x = ± √ 25/4

Having learned how to solve equations of the first degree, of course, you want to work with others, in particular, with equations of the second degree, which are otherwise called quadratic.

Quadratic equations are equations of the type ax ² + bx + c = 0, where the variable is x, the numbers will be - a, b, c, where a is not equal to zero.

If in a quadratic equation one or the other coefficient (c or b) equals zero, then this equation will refer to an incomplete quadratic equation.

How to solve an incomplete quadratic equation if the students have been able to solve only first-degree equations so far? Consider incomplete quadratic equations of different types and simple ways to solve them.

a) If the coefficient c is equal to 0, and the coefficient b is not equal to zero, then ax ² + bx + 0 = 0 is reduced to an equation of the form ax ² + bx = 0.

To solve such an equation, you need to know the formula for solving an incomplete quadratic equation, which consists in factoring the left side of it into factors and later using the condition of equality of the product to zero.

For example, 5x ² - 20x = 0. We factor out the left side of the equation, while performing the usual mathematical operation: taking the common factor out of the brackets

5x (x - 4) = 0

We use the condition that the products are equal to zero.

5 x = 0 or x - 4 = 0

The answer will be: the first root is 0; the second root is 4.

b) If b = 0, and the free term is not equal to zero, then the equation ax ² + 0x + c = 0 is reduced to an equation of the form ax ² + c = 0. The equations are solved in two ways: a) by expanding the polynomial of the equation on the left side into factors ; b) using the properties of the arithmetic square root. Such an equation is solved by one of the methods, for example:

x = ± √ 25/4

x = ± 5/2. The answer is: the first root is 5/2; the second root is - 5/2.

c) If b is equal to 0 and c is equal to 0, then ax ² + 0 + 0 = 0 is reduced to an equation of the form ax ² = 0. In such an equation, x will be equal to 0.

As you can see, incomplete quadratic equations can have no more than two roots.

In a simpler way. To do this, take z out of parentheses. You will get: z (аz + b) = 0. The factors can be written: z = 0 and аz + b = 0, since both can result in zero. In the notation az + b = 0, we move the second one to the right with a different sign. Hence we obtain z1 = 0 and z2 = -b / a. These are the roots of the original.

If there is an incomplete equation of the form аz² + с = 0, in this case they are found by simply transferring the free term to the right side of the equation. Also change its sign when doing this. The result will be az² = -с. Express z² = -c / a. Take the root and write down two solutions - positive and negative square root.

note

If there are fractional coefficients in the equation, multiply the entire equation by the appropriate factor so that you get rid of the fractions.

The knowledge of how to solve quadratic equations is necessary for both schoolchildren and students, sometimes it can also help an adult in everyday life. There are several specific solution methods.

Solving quadratic equations

A quadratic equation of the form a * x ^ 2 + b * x + c = 0. The coefficient x is the desired variable, a, b, c are numerical coefficients. Remember that the "+" sign can change to a "-" sign.

In order to solve this equation, it is necessary to use Vieta's theorem or find the discriminant. The most common way is to find the discriminant, since for some values ​​of a, b, c it is not possible to use Vieta's theorem.

To find the discriminant (D), you need to write the formula D = b ^ 2 - 4 * a * c. The D value can be greater than, less than, or equal to zero. If D is greater or less than zero, then there will be two roots, if D = 0, then only one root remains, more precisely, we can say that D in this case has two equivalent roots. Plug the known coefficients a, b, c into the formula and calculate the value.

After you have found the discriminant, to find x, use the formulas: x (1) = (- b + sqrt (D)) / 2 * a; x (2) = (- b-sqrt (D)) / 2 * a, where sqrt is a function to extract the square root of a given number. By calculating these expressions, you will find two roots of your equation, after which the equation is considered solved.

If D is less than zero, then it still has roots. At school, this section is practically not studied. University students should be aware that a negative number appears at the root. They get rid of it by highlighting the imaginary part, that is, -1 under the root is always equal to the imaginary element "i", which is multiplied by the root with the same positive number. For example, if D = sqrt (-20), after conversion it turns out D = sqrt (20) * i. After this transformation, the solution of the equation is reduced to the same finding of the roots, as described above.

Vieta's theorem is to select the values ​​x (1) and x (2). Two identical equations are used: x (1) + x (2) = -b; x (1) * x (2) = c. Moreover, a very important point is the sign in front of the coefficient b, remember that this sign is opposite to that in the equation. At first glance, it seems that it is very easy to calculate x (1) and x (2), but when solving you will be faced with the fact that the numbers will have to be selected.

Elements for solving quadratic equations

According to the rules of mathematics, some can be decomposed into factors: (a + x (1)) * (b-x (2)) = 0, if you managed to transform this quadratic equation in this way using the formulas of mathematics, then feel free to write down the answer. x (1) and x (2) will be equal to the adjacent coefficients in brackets, but with the opposite sign.

Also, do not forget about incomplete quadratic equations. You may be missing some of the terms, if so, then all its coefficients are simply equal to zero. If there is nothing in front of x ^ 2 or x, then the coefficients a and b are equal to 1.

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 has not been 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 a general quadratic formula. 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, whatever the formula of 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. If the number is negative, 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. The square root expression is the 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 bracket and solve the linear equation, which remains in the brackets. The answer will have two roots. The first one is necessarily equal to zero, because there is a factor consisting of the variable itself. The second is obtained by solving a linear equation.

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 shortcomings are the reason for poor 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 by rewriting them in the standard form: - 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.

Problems for the quadratic equation are studied in the school curriculum and in universities. They are understood as equations of the form a * x ^ 2 + b * x + c = 0, where x - variable, a, b, c - constants; a<>0. The task is to find the roots of the equation.

The geometric meaning of the quadratic equation

The graph of a function that is represented by a quadratic equation is a parabola. The solutions (roots) of the quadratic equation are the points of intersection of the parabola with the abscissa axis (x). It follows that there are three possible cases:
1) the parabola has no points of intersection with the abscissa axis. This means that it is in the upper plane with branches up or lower with branches down. In such cases, the quadratic equation has no real roots (it has two complex roots).

2) the parabola has one point of intersection with the Ox axis. Such a point is called the apex of the parabola, and the quadratic equation in it acquires its minimum or maximum value. In this case, the quadratic equation has one real root (or two identical roots).

3) The last case is more interesting in practice - there are two points of intersection of the parabola with the abscissa axis. This means that there are two real roots of the equation.

Based on the analysis of the coefficients at the degrees of the variables, interesting conclusions can be drawn about the placement of the parabola.

1) If the coefficient a is greater than zero, then the parabola is directed upward, if negative, the parabola branches are directed downward.

2) If the coefficient b is greater than zero, then the vertex of the parabola lies in the left half-plane, if it takes a negative value, then in the right.

Derivation of a formula for solving a quadratic equation

Move the constant from the quadratic equation

for the equal sign, we get the expression

Multiply both sides by 4a

To get a complete square on the left, add b ^ 2 in both parts and carry out the transformation

From here we find

Formula for the discriminant and roots of a quadratic equation

The discriminant is called the value of the radical expression If it is positive then the equation has two real roots, calculated by the formula When the discriminant is zero, the quadratic equation has one solution (two coinciding roots), which can be easily obtained from the above formula when D = 0. When the discriminant is negative, the equation has no real roots. However, solutions of a quadratic equation in the complex plane are found, and their value is calculated by the formula

Vieta's theorem

Consider two roots of a quadratic equation and construct a quadratic equation on their basis. Vieta's theorem follows easily from the notation: if we have a quadratic equation of the form then the sum of its roots is equal to the coefficient p, taken with the opposite sign, and the product of the roots of the equation is equal to the free term q. The formal notation of the above will look like If in the classical equation the constant a is nonzero, then you need to divide the whole equation by it, and then apply Vieta's theorem.

Schedule a quadratic equation for factors

Let the task be set: to factor out a quadratic equation. To perform it, we first solve the equation (find the roots). Next, we substitute the found roots into the expansion formula for the quadratic equation. This will solve the problem.

Quadratic Equation Problems

Objective 1. Find the roots of a quadratic equation

x ^ 2-26x + 120 = 0.

Solution: We write down the coefficients and substitute them into the discriminant formula

The root of this value is 14, it is easy to find it with a calculator, or remember it with frequent use, however, for convenience, at the end of the article I will give you a list of squares of numbers that can often be found in such tasks.
We substitute the found value into the root formula

and we get

Objective 2. Solve the equation

2x 2 + x-3 = 0.

Solution: We have a complete quadratic equation, write out the coefficients and find the discriminant


Using the well-known formulas, we find the roots of the quadratic equation

Objective 3. Solve the equation

9x 2 -12x + 4 = 0.

Solution: We have a full quadratic equation. Determine the discriminant

We got a case when the roots coincide. We find the values ​​of the roots by the formula

Task 4. Solve the equation

x ^ 2 + x-6 = 0.

Solution: In cases where there are small coefficients at x, it is advisable to apply Vieta's theorem. By its condition, we obtain two equations

From the second condition, we get that the product must be equal to -6. This means that one of the roots is negative. We have the following possible pair of solutions (-3; 2), (3; -2). Taking into account the first condition, we reject the second pair of solutions.
The roots of the equation are equal

Problem 5. Find the lengths of the sides of a rectangle if its perimeter is 18 cm and its area is 77 cm 2.

Solution: Half of the perimeter of the rectangle is the sum of the adjacent sides. Let's denote x - the big side, then 18-x is its smaller side. The area of ​​the rectangle is equal to the product of these lengths:
x (18-x) = 77;
or
x 2 -18x + 77 = 0.
Find the discriminant of the equation

Calculate the roots of the equation

If x = 11, then 18's = 7, on the contrary, it is also true (if x = 7, then 21-x = 9).

Problem 6. Factor the 10x 2 -11x + 3 = 0 square equations.

Solution: We calculate the roots of the equation, for this we find the discriminant

Substitute the found value into the root formula and calculate

We apply the formula for the expansion of a quadratic equation in roots

Expanding the brackets, we obtain an identity.

Quadratic equation with parameter

Example 1. For what values ​​of the parameter a , does the equation (a-3) x 2 + (3-a) x-1/4 = 0 have one root?

Solution: By direct substitution of the value a = 3, we see that it has no solution. Next, we will use the fact that for zero discriminant the equation has one root of multiplicity 2. Let us write out the discriminant

simplify it and equate it to zero

We got a quadratic equation for the parameter a, the solution of which is easy to obtain by Vieta's theorem. The sum of the roots is 7, and their product is 12. By simple enumeration, we establish that the numbers 3,4 will be the roots of the equation. Since we have already rejected the solution a = 3 at the beginning of the calculations, the only correct one will be - a = 4. Thus, for a = 4 the equation has one root.

Example 2. For what values ​​of the parameter a , the equation a (a + 3) x ^ 2 + (2a + 6) x-3a-9 = 0 has more than one root?

Solution: Consider first the singular points, they will be the values ​​a = 0 and a = -3. When a = 0, the equation will be simplified to the form 6x-9 = 0; x = 3/2 and there will be one root. For a = -3 we get the identity 0 = 0.
We calculate the discriminant

and find the values ​​of a at which it is positive

From the first condition, we get a> 3. For the second, we find the discriminant and roots of the equation


Let's define the intervals where the function takes positive values. By substituting the point a = 0, we obtain 3>0 . So, outside the interval (-3; 1/3), the function is negative. Don't forget the point a = 0, which should be excluded, since the original equation has one root in it.
As a result, we get two intervals that satisfy the condition of the problem

There will be many similar tasks in practice, try to figure out the tasks yourself and do not forget to take into account the conditions that are mutually exclusive. Learn the formulas for solving quadratic equations well, they are often needed in calculations in various problems and sciences.