Quadratic equations. Discriminant. Solution, examples.
Attention!
There are additional
material in Special Section 555.
For those who strongly "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 keyword is "square". It means that in the equation necessarily there must be an x squared. In addition to it, in the equation there may be (or may not be!) Just x (to the first degree) and just a number (free member). And there should not be x's in a degree 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 but 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 coefficient b and free member of
Such quadratic equations are called complete.
And if b= 0, what will we get? We have X will disappear in the first degree. This happens from multiplying 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, then it is even simpler:
2x 2 \u003d 0,
-0.3x 2 \u003d 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 zero? And you substitute instead a zero.) The X in the square will disappear! The equation will become linear. And it's done differently...
Here are all the main types quadratic equations. Complete and incomplete.
Solution of quadratic equations.
Solution of 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 the standard form, i.e. to the view:
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 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. Here we write:
Example almost solved:
This is the answer.
Everything is very simple. And what do you think, you can't go wrong? Well, yes, how...
The most common mistakes are confusion with the signs of values a, b and c. Or rather, not with their signs (where is there to be confused?), But with the substitution of negative values into the formula for calculating the roots. Here, a detailed record of the formula with specific numbers saves. If there are problems with calculations, so do it!
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 30 seconds to write an extra line. And the number of errors will drop sharply. So we write in detail, with all the brackets and signs:
It seems incredibly difficult to paint so carefully. But it only seems. 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 just turn out right. Especially if you apply practical techniques, which are described below. This evil example with a bunch of minuses will be solved easily and without errors!
But, often, quadratic equations look slightly different. For example, like this:
Did you know?) Yes! This is incomplete quadratic equations.
Solution of incomplete quadratic equations.
They can also be solved by the general formula. You just need to correctly figure out what is equal here a, b and c.
Realized? In the first example a = 1; b = -4; a c? It doesn't exist at all! Well, yes, that's right. In mathematics, this means that c = 0 ! That's all. Substitute zero into the formula instead of c, and everything will work out for us. Similarly with the second example. Only zero we don't have here with, a b !
But incomplete quadratic equations can be solved much easier. Without any formulas. Consider the first not complete equation. What can be done on the left side? You can take the X out of brackets! Let's take it out.
And what from this? And the fact that the product is equal to zero if, and only if any of the factors is equal to zero! Don't believe? Well, then come up with two non-zero numbers that, when multiplied, will give zero!
Does not work? Something...
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 simpler than the general formula. I note, by the way, which X will be the first, and which the second - it is absolutely indifferent. Easy to write in order x 1- whichever is less x 2- that which is more.
The second equation can also be easily solved. Transferring 9 to right side. We get:
It remains to extract the root from 9, and that's it. Get:
also two roots . x 1 = -3, x 2 = 3.
This is how all incomplete quadratic equations are solved. Either by taking X out of brackets, or by simply transferring the number to the right, followed by extracting the root.
It is extremely difficult to confuse these methods. Simply because in the first case you will have to extract the root from X, which is somehow incomprehensible, and in the second case there is nothing to take out of brackets ...
Discriminant. Discriminant formula.
Magic word discriminant ! A rare high school student has not heard this word! The phrase “decide through the discriminant” is reassuring and reassuring. Because there is no need to wait for tricks from the discriminant! It is simple and trouble-free to use.) I remind you of the most general formula for solving any quadratic equations:
The expression under the root sign is called the discriminant. The discriminant is usually denoted by the letter D. Discriminant formula:
D = b 2 - 4ac
And what is so special about this expression? Why does it deserve a special name? What meaning of the discriminant? After all -b, or 2a in this formula they don’t specifically name ... Letters and letters.
The point is this. When solving a quadratic equation using this formula, it is possible only three cases.
1. The discriminant is positive. This means that you can extract the root from it. Whether the root is extracted well or badly is 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 adding or subtracting zero in the numerator does not change anything. Strictly speaking, this is not a single root, but two identical. But, in a simplified version, it is customary to talk about one solution.
3. The discriminant is negative. A negative number does not take the square root. Well, okay. This means there are no solutions.
To be honest, at simple solution quadratic equations, the concept of discriminant is not particularly required. We substitute the values of the coefficients in the formula, and we consider. There everything turns out by itself, and two roots, and one, and not a single one. However, when solving more complex tasks, without knowledge meaning and discriminant formula not enough. Especially - in equations with parameters. Such equations are aerobatics at the GIA and the Unified State Examination!)
So, how to solve quadratic equations through the discriminant you remembered. Or learned, which is also not bad.) You know how to correctly identify a, b and c. Do you know how attentively substitute them into the root formula and attentively count the result. Did you understand that the key word here is - attentively?
Now take note of the practical techniques that dramatically reduce the number of errors. The very ones that are due to inattention ... For which it is then painful and insulting ...
First reception
. Do not be lazy before solving a quadratic equation to bring it to a standard form. What does this mean?
Suppose, after any transformations, you get the following equation:
Do not rush to write the formula of the roots! You will almost certainly mix up the odds a, b and c. Build the example correctly. First, x squared, then without a square, then a free member. Like this:
And again, do not rush! The minus before the x squared can upset you a lot. Forgetting it is easy... Get rid of the minus. How? Yes, as taught in the previous topic! We need to multiply the whole equation by -1. We get:
And now you can safely write down the formula for the roots, calculate the discriminant and complete the example. Decide on your own. You should end up with roots 2 and -1.
Second reception. Check your roots! According to Vieta's theorem. Don't worry, I'll explain everything! Checking last thing the equation. Those. the one by which we wrote down the formula of the roots. If (as in this example) the coefficient a = 1, check the roots easily. It is enough to multiply them. You should get a free term, i.e. in our case -2. Pay attention, not 2, but -2! free member with your sign . If it didn’t work out, it means they already messed up somewhere. Look for an error.
If it worked out, you need to fold the roots. Last and final check. Should be a ratio b with opposite
sign. In our case -1+2 = +1. A coefficient b, which is before the x, is equal to -1. So, everything is correct!
It is a pity that it is so simple only for examples where x squared is pure, with a coefficient a = 1. But at least check in such equations! There will be fewer mistakes.
Reception third . If your equation has fractional coefficients, get rid of the fractions! Multiply the equation by common denominator, as described in the lesson "How to solve equations? Identity transformations". When working with fractions, errors, for some reason, climb ...
By the way, I promised an evil example with a bunch of minuses to simplify. You are welcome! There he is.
In order not to get confused in the minuses, we multiply the equation by -1. We get:
That's all! Deciding is fun!
So let's recap the topic.
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 corresponding factor.
4. If x squared is clean, the coefficient at it 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 \u003d -0.5
x - any number
x 1 = -3
x 2 = 3
no solutions
x 1 = 0.25
x 2 \u003d 0.5
Does everything fit? Fine! Quadratic equations are not your headache. The first three turned out, but the rest did not? Then the problem is not in quadratic equations. The problem is in identical transformations of equations. Take a look at the link, it's helpful.
Doesn't quite work? Or does it not work at all? Then Section 555 will help you. There, all these examples are sorted by bones. Showing main errors in the solution. Of course, it also talks about the use identical 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. 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 to solve equations of the first degree, of course, I 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 ax² + 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) is equal to zero, then this equation will refer to an incomplete quadratic equation.
How to solve an incomplete quadratic equation if students have only been able to solve equations of the first degree so far? Consider incomplete quadratic equations different types and simple ways their decisions.
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 decomposing the left side of it into factors and later using the condition that the product is equal to zero.
For example, 5x ² - 20x \u003d 0. We factor out the left side of the equation, while performing the usual mathematical operation: taking the common factor out of 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 \u003d 0, and the free term is not equal to zero, then the equation ax ² + 0x + c \u003d 0 is reduced to an equation of the form ax ² + c \u003d 0. Solve equations in two ways: a) decomposing 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 ax² + 0 + 0 = 0 reduces to an equation of the form ax² = 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.
More in a simple way. To do this, take z out of brackets. You will get: z(az + b) = 0. Factors can be written: z=0 and az + 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. From here we get z1 = 0 and z2 = -b/a. These are the roots of the original .
If there is an incomplete equation of the form az² + c \u003d 0, in this case they are found by simply transferring the free term to the right side of the equation. Also change its sign. You get the record az² \u003d -s. Express z² = -c/a. Take the root and write down two solutions - a positive and a negative value of the square root.
note
If there are fractional coefficients in the equation, multiply the entire equation by the appropriate factor so as to get rid of the fractions.
Knowledge of how to solve quadratic equations is necessary for both schoolchildren and students, sometimes it can help an adult in ordinary life. There are several specific decision methods.
Solving quadratic equations
A quadratic equation of the form a*x^2+b*x+c=0. Coefficient x is the desired variable, a, b, c - numerical coefficients. Remember that the "+" sign can change to the "-" sign.In order to solve this equation, you must use the Vieta 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 the Vieta theorem.
To find the discriminant (D), you must write the formula D=b^2 - 4*a*c. The value of D 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. Substitute the known coefficients a, b, c into the formula and calculate the value.
After you have found the discriminant, use the formulas to find x: x(1) = (- b+sqrt(D))/2*a; x(2) = (- b-sqrt(D))/2*a where sqrt is the function to take the square root of the given number. After calculating these expressions, you will find the 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 under the root. We get rid of it by separating 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 the transformation, D=sqrt(20)*i is obtained. After this transformation, the solution of the equation is reduced to the same finding of the roots, as described above.
Vieta's theorem consists in the selection of x(1) and x(2) values. Two identical equations are used: x(1) + x(2)= -b; x(1)*x(2)=s. And very important point is the sign before the coefficient b, remember that this sign is the opposite of the one in the equation. At first glance, it seems that calculating x(1) and x(2) is very simple, but when solving, you will encounter the fact that the numbers will have to be selected exactly.
Elements for solving quadratic equations
According to the rules of mathematics, some can be factored: (a + x (1)) * (b-x (2)) \u003d 0, if you managed to transform this quadratic equation in this way using mathematical formulas, 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 x^2 or x is preceded by nothing, then the coefficients a and b are equal to 1.
This topic may seem complicated at first due to the many not-so-simple formulas. Not only do the quadratic equations themselves have long entries, but the roots are also found through the discriminant. There are three new formulas in total. Not very easy to remember. This is possible only after the frequent solution of such equations. Then all the formulas will be remembered by themselves.
General view of the quadratic equation
Their explicit notation is proposed here, when the most major degree listed first, and then in descending order. Often there are situations when the terms stand apart. Then it is better to rewrite the equation in descending order of the degree of the variable.
Let us introduce notation. They are presented in the table below.
If we accept these notations, all quadratic equations are reduced to the following notation.
Moreover, the coefficient a ≠ 0. Let this formula be denoted by number one.
When the equation is given, it is not clear how many roots will be in the answer. Because one of three options is always possible:
- the solution will have two roots;
- the answer will be one number;
- The equation has no roots at all.
And while the decision is not brought to the end, it is difficult to understand which of the options will fall out in a particular case.
Types of records of quadratic equations
Tasks may have different entries. They will not always look like the general formula of a quadratic equation. Sometimes it will lack some terms. What was written above is the complete equation. If you remove the second or third term in it, you get something else. These records are also called quadratic equations, only incomplete.
Moreover, only the terms for which the coefficients "b" and "c" can disappear. The number "a" cannot be equal to zero under any circumstances. Because in this case the formula turns into a linear equation. The formulas for the incomplete form of the equations will be as follows:
So, there are only two types, in addition to complete ones, there are also incomplete quadratic equations. Let the first formula be number two, and the second number three.
The discriminant and the dependence of the number of roots on its value
This number must be known in order to calculate the roots of the equation. It can always be calculated, no matter what the formula of the quadratic equation is. In order to calculate the discriminant, you need to use the equality written below, which will have the number four.
After substituting the values of the coefficients into this formula, you can get numbers with different signs. If the answer is yes, then the answer to the equation will be two different roots. At negative number the roots of the quadratic equation will be missing. If it is equal to zero, the answer will be one.
How is a complete quadratic equation solved?
In fact, consideration of this issue has already begun. Because first you need to find the discriminant. After it is clarified that there are roots of the quadratic equation, and their number is known, you need to use the formulas for the variables. If there are two roots, then you need to apply such a formula.
Since it contains the “±” sign, there will be two values. The expression under the square root sign is the discriminant. Therefore, the formula can be rewritten in a different way.
Formula five. From the same record it can be seen that if the discriminant is zero, then both roots will take the same values.
If the solution of quadratic equations has not yet been worked out, then it is better to write down the values of all coefficients before applying the discriminant and variable formulas. Later this moment will not cause difficulties. But at the very beginning there is confusion.
How is an incomplete quadratic equation solved?
Everything is much simpler here. Even there is no need for additional formulas. And you won't need those that have already been written for the discriminant and the unknown.
First, consider the incomplete equation number two. In this equality, it is supposed to take the unknown value out of the bracket and solve the linear equation, which will remain in the brackets. The answer will have two roots. The first one is necessarily equal to zero, because there is a factor consisting of the variable itself. The second is obtained by solving a linear equation.
The incomplete equation at number three is solved by transferring the number from the left side of the equation to the right. Then you need to divide by the coefficient in front of the unknown. It remains only to extract the square root and do not forget to write it down twice with opposite signs.
The following are some actions that help you learn how to solve all kinds of equalities that turn into quadratic equations. They will help the student to avoid mistakes due to inattention. These shortcomings are the reason bad grades when studying the extensive topic "Quadricular Equations (Grade 8)". Subsequently, these actions will not need to be constantly performed. Because there will be a stable habit.
- First you need to write the equation in standard form. That is, first the term with the largest degree of the variable, and then - without the degree and the last - just a number.
- If a minus appears before the coefficient "a", then it can complicate the work for a beginner to study quadratic equations. It's better to get rid of it. For this purpose, all equality must be multiplied by "-1". This means that all terms will change sign to the opposite.
- In the same way, it is recommended to get rid of fractions. Simply multiply the equation by the appropriate factor so that the denominators cancel out.
Examples
It is required to solve the following quadratic equations:
x 2 - 7x \u003d 0;
15 - 2x - x 2 \u003d 0;
x 2 + 8 + 3x = 0;
12x + x 2 + 36 = 0;
(x+1) 2 + x + 1 = (x+1)(x+2).
The first equation: x 2 - 7x \u003d 0. It is incomplete, therefore it is solved as described for formula number two.
After bracketing, it turns out: x (x - 7) \u003d 0.
The first root takes on the value: x 1 \u003d 0. The second will be found from the linear equation: x - 7 \u003d 0. It is easy to see that x 2 \u003d 7.
Second equation: 5x2 + 30 = 0. Again incomplete. Only it is solved as described for the third formula.
After transferring 30 to the right side of the equation: 5x 2 = 30. Now you need to divide by 5. It turns out: x 2 = 6. The answers will be numbers: x 1 = √6, x 2 = - √6.
Third equation: 15 - 2x - x 2 \u003d 0. Here and below, the solution of quadratic equations will begin by rewriting them in 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 \u003d 0. According to the fourth formula, you need to calculate the discriminant: D \u003d 2 2 - 4 * (- 15) \u003d 4 + 60 \u003d 64. It represents positive number. From what was said above, it turns out that the equation has two roots. They need to be calculated according to the fifth formula. According to it, it turns out that x \u003d (-2 ± √64) / 2 \u003d (-2 ± 8) / 2. Then x 1 \u003d 3, x 2 \u003d - 5.
The fourth equation x 2 + 8 + 3x \u003d 0 is converted to this: x 2 + 3x + 8 \u003d 0. Its discriminant is equal to this value: -23. Since this number is negative, the answer to this task will be the following entry: "There are no roots."
The fifth equation 12x + x 2 + 36 = 0 should be rewritten as follows: x 2 + 12x + 36 = 0. After applying the formula for the discriminant, the number zero is obtained. This means that it will have one root, namely: x \u003d -12 / (2 * 1) \u003d -6.
The sixth equation (x + 1) 2 + x + 1 = (x + 1) (x + 2) requires transformations, which are to bring like terms, before opening the brackets. In place of the first one there will be such an expression: x 2 + 2x + 1. After equality, this entry will appear: x 2 + 3x + 2. After similar terms are counted, the equation will take the form: x 2 - x \u003d 0. It has become incomplete . Similar to it has already been considered a little higher. The roots of this will be the numbers 0 and 1.
Problems on the quadratic equation are also studied in school curriculum and in universities. They are understood as equations of the form a * x ^ 2 + b * x + c \u003d 0, where x- variable, a,b,c – constants; a<>0 . The problem 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 a quadratic equation are the points of intersection of the parabola with the x-axis. It follows that there are three possible cases:
1) the parabola has no points of intersection with the x-axis. This means that it is in the upper plane with branches up or the lower one 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 axis Ox. Such a point is called the vertex 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 powers of the variables, interesting conclusions can be drawn about the placement of the parabola.
1) If the coefficient a Above zero then the parabola is directed with branches upwards, if negative - the branches of the parabola are directed downwards.
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
Let's transfer the constant from the quadratic equation 
for the equal sign, we get the expression
Multiply both sides by 4a 
To get a full square on the left, add b ^ 2 in both parts and perform the transformation
From here we find
Formula of the discriminant and roots of the quadratic equation
The discriminant is 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 are easy to obtain from the above formula for D=0. When the discriminant is negative, there are no real roots. However, to study the solutions of the quadratic equation in the complex plane, 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. The Vieta theorem itself easily follows 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 formula for the above will look like If the constant a in the classical equation is nonzero, then you need to divide the entire equation by it, and then apply the Vieta theorem.
Schedule of the quadratic equation on factors
Let the task be set: to decompose the quadratic equation into factors. To perform it, we first solve the equation (find the roots). Next, we substitute the found roots into the formula for expanding the quadratic equation. This problem will be solved.
Tasks for a quadratic equation
Task 1. Find the roots of a quadratic equation
x^2-26x+120=0 .
Solution: Write down the coefficients and substitute in the discriminant formula
root of given value equal to 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.
The found value is substituted into the root formula 
and we get 
Task 2. solve the equation
2x2+x-3=0.
Solution: We have a complete quadratic equation, write out the coefficients and find the discriminant 
Using well-known formulas, we find the roots of the quadratic equation 
Task 3. solve the equation
9x2 -12x+4=0.
Solution: We have a complete quadratic equation. Determine the discriminant
We got the 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 for x, it is advisable to apply the Vieta 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
Task 5. Find the lengths of the sides of a rectangle if its perimeter is 18 cm and area is 77 cm 2.
Solution: Half the perimeter of a rectangle is equal to the sum of the adjacent sides. Let's denote x - big side, then 18-x is its smaller side. The area of a rectangle is equal to the product of these lengths:
x(18x)=77;
or
x 2 -18x + 77 \u003d 0.
Find the discriminant of the equation
We calculate the roots of the equation 
If a x=11, then 18x=7 , vice versa is also true (if x=7, then 21-x=9).
Problem 6. Factorize the quadratic 10x 2 -11x+3=0 equation.
Solution: Calculate the roots of the equation, for this we find the discriminant 
We substitute the found value into the formula of the roots and calculate 
We apply the formula for expanding the quadratic equation in terms of roots 
Expanding the brackets, we get the 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 \u003d 0 have one root?
Solution: By direct substitution of the value a=3, we see that it has no solution. Further, we will use the fact that with a zero discriminant, the equation has one root of multiplicity 2. Let's write out the discriminant 
simplify it and equate to zero
We have obtained a quadratic equation with respect to the parameter a, the solution of which is easy to obtain using the Vieta 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 .
Calculate the discriminant 
and find the values of a for which it is positive 
From the first condition we get a>3. For the second, we find the discriminant and the roots of the equation 
Let's define the intervals where the function takes positive values. By substituting the point a=0 we get 3>0
.
So, outside the interval (-3; 1/3) the function is negative. Don't forget the dot a=0 which should be excluded, since the original equation in it has one root.
As a result, we obtain two intervals that satisfy the condition of the problem 
There will be many similar tasks in practice, try to deal with the tasks yourself and do not forget to take into account conditions that are mutually exclusive. Study well the formulas for solving quadratic equations, they are quite often needed in calculations in various problems and sciences.