Which is always square. Solution of quadratic equations, formula of roots, examples
The transformation of a complete quadratic equation into an incomplete one looks like this (for the case \(b=0\)):
For cases when \(c=0\) or when both coefficients are equal to zero, everything is similar.
Please note that \(a\) is not equal to zero, it cannot be equal to zero, since in this case it turns into:
Solving incomplete quadratic equations
First of all, you need to understand that the incomplete quadratic equation is still, therefore, it can be solved in the same way as the usual quadratic (through). To do this, we simply add the missing component of the equation with a zero coefficient.
Example
: Find the roots of the equation \(3x^2-27=0\)
Decision
:
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We have an incomplete quadratic equation with the coefficient \(b=0\). That is, we can write the equation in the following form: |
||
|
\(3x^2+0\cdot x-27=0\) |
In fact, here is the same equation as at the beginning, but now it can be solved as an ordinary square. First we write down the coefficients. |
|
|
\(a=3;\) \(b=0;\) \(c=-27;\) |
Calculate the discriminant using the formula \(D=b^2-4ac\) |
|
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\(D=0^2-4\cdot3\cdot(-27)=\) |
Let's find the roots of the equation using the formulas |
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\(x_(1)=\) \(\frac(-0+\sqrt(324))(2\cdot3)\)\(=\)\(\frac(18)(6)\) \(=3\) \(x_(2)=\) \(\frac(-0-\sqrt(324))(2\cdot3)\)\(=\)\(\frac(-18)(6)\) \(=-3\) |
|
Write down the answer |
Answer : \(x_(1)=3\); \(x_(2)=-3\)
Example
: Find the roots of the equation \(-x^2+x=0\)
Decision
:
|
Again, an incomplete quadratic equation, but now the coefficient \(c\) is equal to zero. We write the equation as complete. |
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Quadratic equation or an equation of the second degree with one unknown is an equation that, after transformations, can be reduced to the following form:
ax 2 + bx + c = 0 - quadratic equation
where x is the unknown, and a, b and c- coefficients of the equation. In quadratic equations a is called the first coefficient ( a ≠ 0), b is called the second coefficient, and c is called a known or free member.
The equation:
ax 2 + bx + c = 0
called complete quadratic equation. If one of the coefficients b or c is zero, or both of these coefficients are equal to zero, then the equation is represented as an incomplete quadratic equation.
Reduced quadratic equation
The complete quadratic equation can be reduced to a more convenient form by dividing all its terms by a, that is, for the first coefficient:
The equation x 2 + px + q= 0 is called a reduced quadratic equation. Therefore, any quadratic equation in which the first coefficient is equal to 1 can be called reduced.
For example, the equation:
x 2 + 10x - 5 = 0
is reduced, and the equation:
3x 2 + 9x - 12 = 0
can be replaced by the above equation by dividing all its terms by -3:
x 2 - 3x + 4 = 0
Solving quadratic equations
To solve a quadratic equation, you need to bring it to one of the following forms:
ax 2 + bx + c = 0
ax 2 + 2kx + c = 0
x 2 + px + q = 0
Each type of equation has its own formula for finding the roots:
Pay attention to the equation:
ax 2 + 2kx + c = 0
this is the converted equation ax 2 + bx + c= 0, in which the coefficient b- even, which allows it to be replaced by type 2 k. Therefore, the formula for finding the roots for this equation can be simplified by substituting 2 k instead of b:
Example 1 Solve the equation:
3x 2 + 7x + 2 = 0
Since the second coefficient in the equation is not an even number, and the first coefficient is not equal to one, we will look for the roots using the very first formula, called the general formula for finding the roots of a quadratic equation. At first
a = 3, b = 7, c = 2
Now, to find the roots of the equation, we simply substitute the values of the coefficients into the formula:
| x 1 = | -2 | = - | 1 | , x 2 = | -12 | = -2 |
| 6 | 3 | 6 |
| Answer: - | 1 | , -2. |
| 3 |
Example 2:
x 2 - 4x - 60 = 0
Let's determine what the coefficients are equal to:
a = 1, b = -4, c = -60
Since the second coefficient in the equation is an even number, we will use the formula for quadratic equations with an even second coefficient:
x 1 = 2 + 8 = 10, x 2 = 2 - 8 = -6
Answer: 10, -6.
Example 3
y 2 + 11y = y - 25
Let's bring the equation to a general form:
y 2 + 11y = y - 25
y 2 + 11y - y + 25 = 0
y 2 + 10y + 25 = 0
Let's determine what the coefficients are equal to:
a = 1, p = 10, q = 25
Since the first coefficient is equal to 1, we will look for the roots using the formula for the above equations with an even second coefficient:
Answer: -5.
Example 4
x 2 - 7x + 6 = 0
Let's determine what the coefficients are equal to:
a = 1, p = -7, q = 6
Since the first coefficient is equal to 1, we will look for the roots using the formula for the given equations with an odd second coefficient:
x 1 = (7 + 5) : 2 = 6, x 2 = (7 - 5) : 2 = 1
The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Equations have been used by man since ancient times and since then their use has only increased. The discriminant allows you to solve any quadratic equations using the general formula, which has the following form:
The discriminant formula depends on the degree of the polynomial. The above formula is suitable for solving quadratic equations of the following form:
The discriminant has the following properties that you need to know:
* "D" is 0 when the polynomial has multiple roots (equal roots);
* "D" is a symmetric polynomial with respect to the roots of the polynomial and therefore is a polynomial in its coefficients; moreover, the coefficients of this polynomial are integers, regardless of the extension in which the roots are taken.
Suppose we are given a quadratic equation of the following form:
1 equation
According to the formula we have:
Since \, then the equation has 2 roots. Let's define them:
Where can I solve the equation through the discriminant online solver?
You can solve the equation on our website https: // site. Free online solver will allow you to solve an online equation of any complexity in seconds. All you have to do is just enter your data into the solver. You can also watch the video instruction and learn how to solve the equation on our website. And if you have any questions, you can ask them in our Vkontakte group http://vk.com/pocketteacher. Join our group, we are always happy to help you.
I hope that after studying this article, you will learn how to find the roots of a complete quadratic equation.
With the help of 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 is equations of the form ax 2 + b x + c = 0, where the coefficients a, b and c are not equal to zero. So, to solve the complete quadratic equation, you need to calculate the discriminant D.
D \u003d b 2 - 4ac.
Depending on what value the discriminant has, we will write down the answer.
If the discriminant is a negative number (D< 0),то корней нет.
If the discriminant is zero, then x \u003d (-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 \u003d 4 2 - 4 4 \u003d 0
x = (- (-4))/2 = 2
Answer: 2.
Solve Equation 2 x 2 + x + 3 = 0.
D \u003d 1 2 - 4 2 3 \u003d - 23
Answer: no roots.
Solve Equation 2 x 2 + 5x - 7 = 0.
D \u003d 5 2 - 4 2 (-7) \u003d 81
x 1 \u003d (-5 - √81) / (2 2) \u003d (-5 - 9) / 4 \u003d - 3.5
x 2 \u003d (-5 + √81) / (2 2) \u003d (-5 + 9) / 4 \u003d 1
Answer: - 3.5; one.
So let's imagine the solution of complete quadratic equations by the scheme in Figure 1.
These formulas can be used to solve any complete quadratic equation. You just need to be careful to the equation was written as a polynomial of standard form
a x 2 + bx + c, otherwise you can 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 \u003d 3 2 - 4 1 2 \u003d 1 and then the equation has two roots. And this is not true. (See example 2 solution 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 be in the first place, that is a x 2 , then with less – bx, and then the free term with.
When solving the above quadratic equation and the quadratic equation with an even coefficient for the second term, other formulas can also be used. Let's get acquainted with these formulas. If in the full quadratic equation with the second term the coefficient is even (b = 2k), then the equation can be solved using the formulas shown in the diagram of Figure 2.
A complete quadratic equation is called reduced if the coefficient at x 2 equals unity and the equation takes the form x 2 + px + q = 0. Such an equation can be given to solve, or is obtained by dividing all the coefficients of the equation by the coefficient a standing at x 2 .
Figure 3 shows a diagram of the solution of the reduced square 
equations. Consider the 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 Figure 1.
D \u003d 6 2 - 4 3 (- 6) \u003d 36 + 72 \u003d 108
√D = √108 = √(36 3) = 6√3
x 1 \u003d (-6 - 6 √ 3) / (2 3) \u003d (6 (-1- √ (3))) / 6 \u003d -1 - √ 3
x 2 \u003d (-6 + 6 √ 3) / (2 3) \u003d (6 (-1 + √ (3))) / 6 \u003d -1 + √ 3
Answer: -1 - √3; –1 + √3
You can see that the coefficient at x in this equation is an even number, that is, b \u003d 6 or b \u003d 2k, whence k \u003d 3. Then let's try to solve the equation using the formulas shown in the figure diagram D 1 \u003d 3 2 - 3 (- 6 ) = 9 + 18 = 27
√(D 1) = √27 = √(9 3) = 3√3
x 1 \u003d (-3 - 3√3) / 3 \u003d (3 (-1 - √ (3))) / 3 \u003d - 1 - √3
x 2 \u003d (-3 + 3√3) / 3 \u003d (3 (-1 + √ (3))) / 3 \u003d - 1 + √3
Answer: -1 - √3; –1 + √3. Noticing that all the coefficients in this quadratic equation are divisible by 3 and dividing, we get the reduced quadratic equation x 2 + 2x - 2 = 0 We solve this equation using the formulas for the reduced quadratic 
equations figure 3.
D 2 \u003d 2 2 - 4 (- 2) \u003d 4 + 8 \u003d 12
√(D 2) = √12 = √(4 3) = 2√3
x 1 \u003d (-2 - 2√3) / 2 \u003d (2 (-1 - √ (3))) / 2 \u003d - 1 - √3
x 2 \u003d (-2 + 2 √ 3) / 2 \u003d (2 (-1 + √ (3))) / 2 \u003d - 1 + √ 3
Answer: -1 - √3; –1 + √3.
As you can see, when solving this equation using different formulas, we got the same answer. Therefore, having well mastered the formulas shown in the diagram of Figure 1, you can always solve any complete quadratic equation.
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Lesson contentWhat is a quadratic equation and how to solve it?
We remember that an equation is an equality containing a variable whose value needs to be found.
If the variable included in the equation is raised to the second power (squared), then such an equation is called second degree equation or quadratic equation.
For example, the following equations are quadratic:
We solve the first of these equations, namely x 2 − 4 = 0 .
All the identical transformations that we used in solving ordinary linear equations can also be applied in solving square ones.
So in the equation x 2 − 4 = 0 we move the term −4 from the left side to the right side by changing the sign:
Got the equation x 2 = 4 . Earlier we said that the equation is considered solved if in one part the variable is written in the first degree and its coefficient is equal to one, and the other part is equal to some number. That is, to solve the equation, it should be reduced to the form x = a, where a- root of the equation.
We have a variable x still in the second degree, so the solution must be continued.
To solve the equation x 2 = 4 , you need to answer the question at what value x the left side becomes 4 . Obviously, for the values 2 and −2 . To derive these values, we use the definition of the square root.
Number b called the square root of the number a, if b 2 = a and is denoted as
We have a similar situation now. After all, what is x 2 = 4? Variable x in this case it is the square root of 4, because the second power x equated to 4.
Then we can write that . The calculation of the right side will allow you to find out what is equal to x. The square root has two meanings: positive and negative. Then we get x= 2 and x= −2 .
Usually they write it like this: they put a plus-minus sign in front of the square root, then find. In our case, at the stage when the expression is written, the ± sign should be placed before
Then find the arithmetic value of the square root
Expression x= ± 2 means that x= 2 and x= −2 . That is, the roots of the equation x 2 − 4 = 0 are the numbers 2 and −2 . We write the complete solution of this equation:
In both cases, the left side is zero. So the equation is correct.
Let's solve another equation. Let it be required to solve the quadratic equation ( x+ 2) 2 = 25
First, let's analyze this equation. The left side is squared and equals 25 . What number squared is 25? Obviously, the numbers 5 and −5
That is, our task is to find x, under which the expression x+ 2 will be equal to the numbers 5 and −5 . Let's write these two equations:
Let's solve both equations. These are ordinary linear equations that are easily solved:
So the roots of the equation ( x+ 2) 2 = 25 are the numbers 3 and −7 .
In this example, as in the past, you can use the definition of the square root. So, in the equations ( x+ 2) 2 = 25 expression ( x+ 2) is the square root of 25. Therefore, we can first write that 
.
Then the right side becomes equal to ±5 . You get two equations: x+ 2 = 5 and x+ 2 = −5. By solving each of these equations separately, we will come to the roots 3 and −7.
Let us write down the complete solution of the equation ( x+ 2) 2 = 25
From the examples considered, it can be seen that the quadratic equation has two roots. In order not to forget about the found roots, the variable x can be signed with subscripts. Thus, the root 3 can be denoted as x 1 , and the root −7 through x 2
In the previous example, you could also do this. The equation x 2 − 4 = 0 had roots 2 and −2 . These roots can be referred to as x 1 = 2 and x 2 = −2.
It also happens that a quadratic equation has only one root or no roots at all. We will consider such equations later.
Let's check the equation ( x+ 2) 2 = 25 . Substitute the roots 3 and −7 into it. If for values 3 and −7 the left side is equal to 25 , then this will mean that the equation is solved correctly:
In both cases, the left side is 25 . So the equation is correct.
The quadratic equation is given in different forms. Its most common form looks like this:
ax 2 + bx + c= 0
,
where a, b, c- some numbers x- unknown.
This so-called general form of the quadratic equation. In such an equation, all terms are collected in a common place (in one part), and the other part is equal to zero. Otherwise, this type of equation is called normal form of a quadratic equation.
Let equation 3 be given x 2 + 2x= 16 . It has a variable x raised to the second power, so the equation is quadratic. Let us bring this equation to a general form.
So, we need to get an equation that will be similar to the equation ax 2 + bx+ c= 0 . For this, in equation 3 x 2 + 2x= 16 we move 16 from the right side to the left side by changing the sign:
3x 2 + 2x − 16 = 0
Got the equation 3x 2 + 2x− 16 = 0 . In this equation a= 3 , b= 2 , c= −16 .
In a quadratic equation of the form ax 2 + bx+ c= 0 numbers a , b and c have their own names. Yes, the number a called the first or senior coefficient; number b called the second coefficient; number c called a free member.
In our case, for the equation 3x 2 + 2x− 16 = 0 the first or highest coefficient is 3 ; the second coefficient is the number 2 ; the free member is the number −16 . There is another common name for numbers a, b and c — options.
So, in the equation 3x 2 + 2x− 16 = 0 the parameters are the numbers 3 , 2 and −16 .
In a quadratic equation, it is desirable to arrange the terms so that they are arranged in the same order as in the normal form of the quadratic equation.
For example, given the equation −5 + 4x 2 + x= 0 , then it is desirable to write it in the normal form, that is, in the form ax 2 + bx + c= 0.
In the equation −5 + 4x 2 + x = 0 it can be seen that the free term is -5 , it should be located at the end of the left side. Member 4 x 2 contains the leading coefficient, it must be placed first. Member x respectively will be located second:
The quadratic equation can take different forms depending on the case. It all depends on what the values are a , b and with .
If the coefficients a , b and c are not equal to zero, then the quadratic equation is called complete. For example, the quadratic equation is complete 2x 2 + 6x - 8 = 0 .
If any of the coefficients is equal to zero (that is, absent), then the equation is significantly reduced and takes on a simpler form. This quadratic equation is called incomplete. For example, the quadratic equation 2 is incomplete x 2 + 6x= 0, it has coefficients a and b(numbers 2 and 6 ), but there is no free member c.
Let's consider each of these types of equations, and for each of these types we will define its own way of solving.
Let the quadratic equation 2x 2 + 6x - 8 = 0 . In this equation a= 2 , b= 6 , c= −8 . If a b set equal to zero, then the equation will take the form:
It turned out equation 2 x 2 − 8 = 0 . To solve it, we move −8 to the right side, changing the sign:
2x 2 = 8
To further simplify the equation, we use the previously studied identical transformations. In this case, you can divide both parts into 2
We have the equation that we solved at the beginning of this lesson. To solve the equation x 2 \u003d 4, you should use the definition of the square root. If a x 2 = 4 , then . From here x= 2 and x= −2 .
So the roots of equation 2 x 2 − 8 = 0 are the numbers 2 and −2 . We write the complete solution of this equation:
Let's do a check. We substitute the roots 2 and −2 into the original equation and perform the corresponding calculations. If for the values 2 and −2 the left side is equal to zero, then this will mean that the equation is solved correctly:
In both cases, the left side is equal to zero, which means the equation is solved correctly.
The equation we have now solved is incomplete quadratic equation. The name speaks for itself. If the complete quadratic equation looks like ax 2 + bx+ c= 0 , then making the coefficient b zero is an incomplete quadratic equation ax 2 + c= 0 .
We also first had a complete quadratic equation 2x 2 + 6x− 4 = 0 . But we made the ratio b zero, that is, instead of the number 6 put 0 . As a result, the equation turned into an incomplete quadratic equation 2 x 2 − 4 = 0 .
At the beginning of this lesson, we solved the quadratic equation x 2 − 4 = 0 . It is also an equation of the form ax 2 + c= 0 , that is, incomplete. In him a= 1 , b= 0 , with= −4 .
Also, the quadratic equation will be incomplete if the coefficient c equals zero.
Consider the complete quadratic equation 2x 2 + 6x - 4 = 0 . Let's make a coefficient c zero. That is, instead of the number 4, put 0
Got a quadratic equation 2 x 2 + 6x=0 , which is incomplete. To solve such an equation, the variable x put out of brackets:
It turned out the equation x(2x+ 6) = 0 in which to find x, at which the left side becomes equal to zero. Note that in this equation the expressions x and 2 x+ 6) are factors. One of the properties of multiplication says that the product is equal to zero if at least one of the factors is equal to zero (either the first factor or the second).
In our case, equality will be achieved if x will be equal to zero or (2 x+ 6) will be equal to zero. Let's start by writing this:
There are two equations: x= 0 and 2 x+ 6 = 0 . The first equation does not need to be solved - it has already been solved. That is, the first root is zero.
To find the second root, we solve equation 2 x+ 6 = 0 . This is a simple linear equation that is easy to solve:
We see that the second root is −3.
So the roots of equation 2 x 2 + 6x= 0 are the numbers 0 and −3 . We write the complete solution of this equation:
Let's do a check. We substitute the roots 0 and −3 into the original equation and perform the corresponding calculations. If for the values 0 and −3 the left side is equal to zero, then this will mean that the equation is solved correctly:
The next case is when the numbers b and with are equal to zero. Consider the complete quadratic equation 2x 2 + 6x− 4 = 0 . Let's make coefficients b and c zeros. Then the hello equation is:
Got Equation 2 x 2 = 0 . The left side is the product and the right side is zero. The product is equal to zero if at least one of the factors is equal to zero. It's obvious that x= 0 . Indeed, 2 × 0 2 = 0 . Hence, 0 = 0 . For other values x equality will not be achieved.
Simply put, if in a quadratic equation of the form ax 2 + bx+ c= 0 numbers b and with are equal to zero, then the root of such an equation is equal to zero.
Note that when the phrases " b is zero" or " c is zero “, then it is understood that the parameters b or c not included in the equation at all.
For example, if given equation 2 x 2 − 32 = 0 , then we say that b= 0 . Because when compared to the full equation ax 2 + bx+ c= 0 , it can be seen that in equation 2 x 2 − 32 = 0 there is a leading coefficient a, equal to 2; there is an intercept -32 ; but no coefficient b .
Finally, consider the complete quadratic equation ax 2 + bx+ c= 0 . As an example, let's solve the quadratic equation x 2 − 2x+ 1 = 0 .
So we need to find x, at which the left side becomes equal to zero. Let us use the identical transformations studied earlier.
First of all, note that the left side of the equation is . If we remember how , we get on the left side ( x− 1) 2 .
We argue further. The left side is squared and equals zero. What number squared is zero? Obviously only 0 . Therefore, our task is to find x, at which the expression x− 1 is equal to zero. By solving the simplest equation x− 1 = 0 , you can find out what is equal to x
The same result can be obtained by using the square root. In the equation ( x− 1) 2 = 0 expression ( x− 1) is the square root of zero. Then one can write that 
. In this example, you do not need to write the ± sign before the root, since the root of zero has only one value - zero. Then it turns out x− 1 = 0 . From here x= 1
.
So the root of the equation x 2 − 2x+ 1 = 0 is a unit. This equation has no other roots. In this case, we have solved a quadratic equation that has only one root. This also happens.
Simple equations are not always given. Consider for example the equation x 2 + 2x− 3 = 0 .
In this case, the left side is no longer the square of the sum or difference. Therefore, other solutions must be found.
Note that the left side of the equation is a quadratic trinomial. Then we can try to select a full square from this trinomial and see what it gives us.
We select the full square from the square trinomial located on the left side of the equation:
In the resulting equation, we transfer −4 to the right side by changing the sign:
Now let's use the square root. In the equation ( x+ 1) 2 = 4 expression ( x+ 1) is the square root of 4. Then one can write that 
. Calculating the right side will give the expression x+ 1 = ±2 . From this we get two equations: x+ 1 = 2 and x+ 1 = −2 whose roots are the numbers 1 and −3
So the roots of the equation x 2 + 2x− 3 = 0 are the numbers 1 and −3 .
Let's check:
Example 3. solve the equation x 2 − 6x+ 9 = 0 , selecting a full square.
So the root of the equation x 2 − 6x+ 9 = 0 is 3. Let's check:
Example 4 4x 2 + 28x− 72 = 0 , highlighting the full square:
Select a full square from the left side:
Let's move −121 from the left side to the right side, changing the sign:
Let's use the square root:
We got two simple equations: 2 x+ 7 = 11 and 2 x+ 7 = -11. Let's solve them:
Example 5. solve the equation 2x 2 + 3x− 27 = 0
This equation is a little more complicated. When we select a full square, we represent the first term of the square trinomial as a square of some expression.
So, in the previous example, the first term of the equation was 4 x 2. It could be represented as a square of the expression 2 x, i.e (2x) 2 = 2 2 x 2 = 4x 2 . To verify that this is correct, you can take the square root of the expression 4 x 2. This is the square root of the product - it is equal to the product of the roots:
In the equation 2x 2 + 3x− 27 = 0 first member is 2 x 2. It cannot be represented as a square of any expression. Because there is no number whose square is 2. If there were such a number, then this number would be the square root of the number 2. But the square root of the number 2 is extracted only approximately. And the approximate value is not suitable for representing the number 2 as a square.
If both parts of the original equation are multiplied or divided by the same number, then the equation is equivalent to the original one. This rule holds for the quadratic equation as well.
Then we can divide both sides of our equation by 2. This will get rid of the deuce before x 2 which will later give us the opportunity to select a full square:
Rewrite the left side as three fractions with denominator 2
We reduce the first fraction by 2. We rewrite the remaining members of the left side without changes. The right side will still become zero:
Let's select a full square.
When a term is represented as a double product, the appearance of a factor of 2 would lead to the fact that this factor and the denominator of the fraction would be reduced. To prevent this from happening, the doubled product was multiplied by. When selecting a full square, you should always try to make sure that the value of the original expression does not change.
Let's collapse the resulting full square:
Here are similar members:
Let's move the fraction to the right side by changing the sign:
Let's use the square root. The expression is the square root of a number
To calculate the right side, we use the extraction rule:
Then our equation will take the form:
We get two equations:
Let's solve them:
So the roots of the equation 2x 2 + 3x− 27 = 0 are the numbers 3 and .
It is more convenient to leave the root in this form, without dividing the numerator by the denominator. This will make it easier to check.
Let's do a check. We substitute the found roots into the original equation:
In both cases, the left side is equal to zero, so the equation 2x 2 + 3x− 27 = 0 decided right.
Solving the Equation 2x 2 + 3x− 27 = 0 , at the very beginning we divided both parts of it by 2 . As a result, a quadratic equation was obtained in which the coefficient before x 2 is equal to one:
This kind of quadratic equation is called reduced quadratic equation.
Any quadratic equation of the form ax 2 + bx+ c= 0 can be made reduced. To do this, you need to divide both its parts by the coefficient that is located in front of x². In this case, both sides of the equation ax 2 + bx+ c= 0 needs to be divided into a
Example 6. Solve a quadratic equation 2x 2 + x+ 2 = 0
Let's make this equation reduced:
Let's select a full square:
Got the equation 
, in which the square of the expression is equal to a negative number. This cannot be, since the square of any number or expression is always positive.
Therefore, there is no such x, at which the left side would become equal to . So the equation has no roots.
And since the equation is equivalent to the original equation 2x 2 + x+ 2 = 0
, then it (the original equation) has no roots.
Formulas for the roots of a quadratic equation
Selecting a full square for each quadratic equation being solved is not very convenient.
Is it possible to create universal formulas for solving quadratic equations? It turns out you can. Now we will deal with this.
Based on the literal equation ax 2 + bx+ c= 0 , and after performing some identical transformations, we can get formulas for deriving the roots of the quadratic equation ax 2 + bx+ c= 0 . Coefficients can be substituted into these formulas a , b , with and get solutions.
So, we select the full square from the left side of the equation ax 2 + bx+ c= 0. First, let's make this equation reduced. Let's divide both parts into a
Now, in the resulting equation, we select the full square:
We transfer the terms and to the right side by changing the sign:
Let's bring the right side to a common denominator. Fractions consisting of letters lead to a common denominator. That is, the denominator of the first fraction becomes the additional factor of the second fraction, and the denominator of the second fraction becomes the additional factor of the first fraction:
In the numerator of the right side, we take out of brackets a
Let us shorten the right side by a
Since all transformations were identical, the resulting equation 
has the same roots as the original equation ax 2 + bx+ c= 0.
The equation will have roots only if the right side is greater than or equal to zero. This is because the squaring is done on the left side, and the square of any number is positive or equal to zero (if zero is squared into this square). And what the right side will be equal to depends on what will be substituted instead of variables a
, b and c
.
Because for any a not equal to zero, the denominator of the right side of the equation will always be positive, then the sign of the fraction will depend on the sign of its numerator, that is, on the expression b 2 − 4ac
.
Expression b 2 − 4ac called discriminant of a quadratic equation. Discriminant is a Latin word meaning distinguisher . The discriminant of a quadratic equation is denoted by the letter D
D = b 2 − 4ac
The discriminant allows you to know in advance whether the equation has roots or not. So, in the previous task, we solved the equation for a long time 2x 2 + x+ 2 = 0 and it turned out that it has no roots. The discriminant would allow us to know in advance that there are no roots. In the equation 2x 2 + x+ 2 = 0 odds a , b and c are 2, 1 and 2, respectively. Substitute them into the formula D = b 2 −4ac
D = b 2 − 4ac= 1 2 − 4 × 2 × 2 = 1 − 16 = −15.
We see that D(it is b 2 − 4ac) is a negative number. Then there is no point in solving the equation 2x 2 + x+ 2 = 0,
selecting a full square in it, because when we get to an equation of the form , it turns out that the right side becomes less than zero (due to the negative discriminant). And the square of a number cannot be negative. Therefore, this equation has no roots.
It becomes clear why ancient people considered the expression b 2 − 4ac distinguisher. This expression, like an indicator, allows you to distinguish between an equation with roots and an equation without roots.
So, D equals b 2 − 4ac. Substitute in the equation instead of expression b 2 − 4ac letter D
If the discriminant of the original equation is less than zero ( D< 0) , то уравнение примет вид:
In this case, the original equation is said to have no roots, since the square of any number must not be negative.
If the discriminant of the original equation is greater than zero ( D> 0) , then the equation will take the form:
In this case, the equation will have two roots. To derive them, we use the square root:
Got the equation 
. From it we get two equations: 
and 
. Express x in each of the equations:
The resulting two equalities are the universal formulas for solving the quadratic equation ax 2 + bx+ c= 0. They are called formulas of the roots of the quadratic equation.
Most often, these formulas are denoted as x 1 and x 2. That is, to calculate the first root, a formula with index 1 is used; to derive the second root - a formula with index 2. Let's denote our formulas in the same way:
The order in which the formulas are applied is not important.
Let's solve for example a quadratic equation x 2 + 2x− 8 = 0 using the formulas of the roots of a quadratic equation. The coefficients of this quadratic equation are the numbers 1 , 2 and −8 . I.e, a= 1 , b= 2 , c= −8 .
Before using the formulas for the roots of a quadratic equation, you need to find the discriminant of this equation.
Let's find the discriminant of the quadratic equation. To do this, we use the formula D = b 2 − 4 ac. Instead of variables a, b and c we will have the coefficients of the equation x 2 + 2x− 8 = 0
D = b 2 − 4ac= 2 2 − 4 × 1 × (−8) = 4 + 32 = 36
The discriminant is greater than zero. So the equation has two roots. Now you can use the formulas of the roots of the quadratic equation:
So the roots of the equation x 2 + 2x− 8 = 0 are the numbers 2 and −4 . Checking to make sure that the roots are found correctly:
Finally, consider the case when the discriminant of the quadratic equation is equal to zero. Let's go back to the equation. If the discriminant is zero, then the right side of the equation will take the form:
And in this case, the quadratic equation will have only one root. Let's use the square root:
This is another formula for deriving the square root. Let's consider its application. Earlier we solved the equation x 2 − 6x+ 9 = 0 , which has one root 3. We solved it by selecting a full square. Now let's try to solve using formulas.
Let's find the discriminant of the quadratic equation. In this equation a= 1 , b= −6 , c= 9 . Then, according to the discriminant formula, we have:
D = b 2 − 4ac= (−6) 2 − 4 × 1 × 9 = 36 − 36 = 0
The discriminant is zero ( D= 0) . This means that the equation has only one root, and it is calculated by the formula
So the root of the equation x 2 − 6x+ 9 = 0 is the number 3.
For a quadratic equation that has one root, the formulas are also applicable 
and 
. But applying each of them will give the same result.
Let's apply these two formulas for the previous equation. In both cases we get the same answer 3
If the quadratic equation has only one root, then it is advisable to use the formula, and not the formulas and . This saves time and space.
Example 3. solve the equation 5x 2 − 6x+ 1 = 0
So the roots of the equation 5x 2 − 6x+ 1 = 0 are the numbers 1 and .
Answer: 1; .
Example 4. solve the equation x 2 + 4x+ 4 = 0
Let's find the discriminant of the quadratic equation:
The discriminant is zero. So the equation has only one root. It is calculated according to the formula
So the root of the equation x 2 + 4x+ 4 = 0 is the number −2.
Answer: -2.
Example 5. solve the equation 3x 2 + 2x+ 4 = 0
Let's find the discriminant of the quadratic equation:
The discriminant is less than zero. So this equation has no roots.
Answer: no roots.
Example 6. solve the equation (x+ 4) 2 = 3x+ 40
Let's bring this equation to normal form. On the left side is the square of the sum of two expressions. Let's break it down:
Let's move all the terms from the right side to the left side by changing their signs. Zero will remain on the right side:
The discriminant is greater than zero. So the equation has two roots. Let's use the formulas of the roots of the quadratic equation:
So the roots of the equation (x+ 4) 2 = 3x+ 40 are the numbers 3 and −8 .
Answer: 3; −8.
Example 7. solve the equation 
Multiply both sides of this equation by 2. This will allow us to get rid of the fraction on the left side:
In the resulting equation, we transfer 22 from the right side to the left side by changing the sign. 0 will remain on the right side
Here are similar terms on the left side:
In the resulting equation, we find the discriminant:
The discriminant is greater than zero. So the equation has two roots. Let's use the formulas of the roots of the quadratic equation:
So the roots of the equation are the numbers 23 and −1 .
Answer: 23; −1.
Example 8. solve the equation 
Multiply both parts by the least common multiple of the denominators of both fractions. This will get rid of the fractions in both parts. The least common multiple of 2 and 3 is 6 . Then we get:
In the resulting equation, open the brackets in both parts:
Now let's transfer all the terms from the right side to the left side, changing their signs. 0 will remain on the right side
Here are similar terms on the left side:
In the resulting equation, we find the discriminant:
The discriminant is greater than zero. So the equation has two roots. Let's use the formulas of the roots of the quadratic equation:
So the roots of the equation are numbers and 2.
Examples of solving quadratic equations
Example 1. solve the equation x 2 = 81
This is the simplest quadratic equation in which you need to determine the number whose square is 81. These are the numbers 3 and −3. Let's use the square root to derive them:
Answer: 9, −9 .
Example 2. solve the equation x 2 − 9 = 0
This is an incomplete quadratic equation. To solve it, you need to move the term −9 to the right side by changing the sign. Then we get:
Answer: 3, −3.
Example 3. solve the equation x 2 − 9x= 0
This is an incomplete quadratic equation. To solve it, you first need to take out x for brackets:
The left side of the equation is the product. The product is equal to zero if at least one of the factors is equal to zero.
The left side will become equal to zero if separately x is zero, or if the expression x− 9 is equal to zero. You get two equations, one of which has already been solved:
Answer: 0, 9 .
Example 4. solve the equation x 2 + 4x− 5 = 0
This is a complete quadratic equation. It can be solved by the method of selecting a full square or using the formulas of the roots of a quadratic equation.
Let's solve this equation using formulas. Let's find the discriminant first:
D= b 2 − 4ac= 4 2 − 4 × 1 × (−5) = 16 + 20 = 36
The discriminant is greater than zero. So the equation has two roots. Let's calculate them:
Answer: 1, −5 .
Example 5. solve the equation 
Let's multiply both parts by 5, 3 and 6. This will get rid of the fractions in both parts:
In the resulting equation, we transfer all terms from the right side to the left side by changing the sign. Zero will remain on the right side:
Here are similar members:
Answer: 5 , .
Example 6. solve the equation x 2 = 6
In this example, as you need to use the square root:
However, the square root of 6 is not taken. It is extracted only approximately. The root can be extracted with a certain accuracy. Let's extract it to the nearest hundredth:
But most often the root is left as a radical:
Answer:
Example 7. solve the equation (2x+ 3) 2 + (x− 2) 2 = 13
Let's open the brackets on the left side of the equation:
In the resulting equation, we transfer 13 from the right side to the left side, changing the sign. Then we give similar members:
We got an incomplete quadratic equation. Let's solve it:
Answer: 0 , −1,6 .
Example 8. solve the equation (5 + 7x)(4 − 3x) = 0
This equation can be solved in two ways. Let's consider each of them.
First way. Expand the brackets and get the normal form of the quadratic equation.
Let's expand the brackets:
Here are similar members:
We rewrite the resulting equation so that the term with the highest coefficient is located first, the term with the second coefficient is second, and the free term is located third:
To make the leading term positive, we multiply both sides of the equation by −1. Then all the terms of the equation will change their signs to the opposite:
We solve the resulting equation using the formulas of the roots of the quadratic equation:
Second way. Find values x, for which the factors on the left side of the equation are equal to zero. This method is more convenient and much shorter.
The product is equal to zero if at least one of the factors is equal to zero. In this case, the equality in the equation (5 + 7x)(4 − 3x) = 0 will be achieved if the expression (5 + 7 x) is equal to zero, or the expression (4 − 3 x) is zero. Our task is to find out under what x it happens:
Examples of problem solving
Imagine that it became necessary to build a small room with an area of 8 m 2. In this case, the length of the room should be twice its width. How to determine the length and width of such a room?
Let's make an approximate drawing of this room, which illustrates the top view:
Denote the width of the room through x. And the length of the room after 2 x, because according to the condition of the problem, the length should be twice the width. The multiplier is 2 and will fulfill this requirement:
The surface of the room (its floor) is a rectangle. To calculate the area of a rectangle, multiply the length of the rectangle by its width. Let's do it:
2x × x
According to the condition of the problem, the area should be 8 m 2. So expression 2 x× x should be equated to 8
2x × x = 8
Got an equation. If you solve it, you can find the length and width of the room.
The first thing you can do is do the multiplication on the left side of the equation:
2x 2 = 8
As a result of this transformation, the variable x moved to the second degree. And we said that if the variable included in the equation is raised to the second power (squared), then such an equation is an equation of the second degree or a quadratic equation.
To solve our quadratic equation, we use the previously studied identical transformations. In this case, you can divide both parts into 2
Now let's use the square root. If a x 2 = 4 , then . From here x= 2 and x= −2 .
Through x the width of the room was indicated. The width must not be negative, so only the value 2 is taken into account. This often happens when solving problems in which a quadratic equation is used. Two roots are obtained in the answer, but only one of them satisfies the condition of the problem.
And the length was indicated by 2 x. Meaning x now known, substitute it into expression 2 x and calculate the length:
2x= 2 × 2 = 4
So the length is 4 m, and the width is 2 m. This solution satisfies the condition of the problem, since the area of \u200b\u200bthe room is 8 m 2
4 × 2 = 8 m 2
Answer: The length of the room is 4m and the width is 2m.
Example 2. A garden plot in the shape of a rectangle, one side of which is 10 m longer than the other, needs to be surrounded by a fence. Determine the length of the fence, if it is known that the area of \u200b\u200bthe site is 1200 m 2
Decision
The length of a rectangle is usually greater than its width. Let the plot width x meters, and the length ( x+ 10) meters. The plot area is 1200 m 2 . Multiply the length of the section by its width and equate to 1200, we get the equation:
x(x+ 10) = 1200
Let's solve this equation. First, open the brackets on the left side:
Let's move 1200 from the right side to the left side by changing the sign. 0 will remain on the right side
We solve the resulting equation using the formulas:
Despite the fact that the quadratic equation has two roots, we take into account only the value 30. Because the width cannot be expressed as a negative number.
So through x the width of the area was marked. It is equal to thirty meters. And the length was indicated through the expression x+ 10 . Substitute the found value into it x and calculate the length:
x
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