We continue to study the topic solution of equations". We have already got acquainted with linear equations and now we are going to get acquainted with quadratic equations.
First, we will analyze what a quadratic equation is, how it is written in general view, and give related definitions. After that, using examples, we will analyze in detail how incomplete quadratic equations are solved. Next, let's move on to solving the complete equations, get the formula for the roots, get acquainted with the discriminant quadratic equation and consider solutions characteristic examples. Finally, we trace the connections between roots and coefficients.
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What is a quadratic equation? Their types
First you need to clearly understand what a quadratic equation is. Therefore, it is logical to start talking about quadratic equations with the definition of a quadratic equation, as well as definitions related to it. After that, you can consider the main types of quadratic equations: reduced and non-reduced, as well as complete and incomplete equations.
Definition and examples of quadratic equations
Definition.
Quadratic equation is an equation of the form a x 2 +b x+c=0, where x is a variable, a , b and c are some numbers, and a is different from zero.
Let's say right away that quadratic equations are often called equations of the second degree. This is because the quadratic equation is algebraic equation second degree.
The sounded definition allows us to give examples of quadratic equations. So 2 x 2 +6 x+1=0, 0.2 x 2 +2.5 x+0.03=0, etc. are quadratic equations.
Definition.
Numbers a , b and c are called coefficients of the quadratic equation a x 2 + b x + c \u003d 0, and the coefficient a is called the first, or senior, or coefficient at x 2, b is the second coefficient, or coefficient at x, and c is a free member.
For example, let's take a quadratic equation of the form 5 x 2 −2 x−3=0, here the leading coefficient is 5, the second coefficient is −2, and the free term is −3. Note that when the coefficients b and/or c are negative, as in the example just given, the short form of the quadratic equation of the form 5 x 2 −2 x−3=0 is used, not 5 x 2 +(−2 )x+(−3)=0 .
It is worth noting that when the coefficients a and / or b are equal to 1 or −1, then they are usually not explicitly present in the notation of the quadratic equation, which is due to the peculiarities of the notation of such . For example, in the quadratic equation y 2 −y+3=0, the leading coefficient is one, and the coefficient at y is −1.
Reduced and non-reduced quadratic equations
Depending on the value of the leading coefficient, reduced and non-reduced quadratic equations are distinguished. Let us give the corresponding definitions.
Definition.
A quadratic equation in which the leading coefficient is 1 is called reduced quadratic equation. Otherwise, the quadratic equation is unreduced.
According to this definition, quadratic equations x 2 −3 x+1=0 , x 2 −x−2/3=0, etc. - reduced, in each of them the first coefficient is equal to one. And 5 x 2 −x−1=0 , etc. - unreduced quadratic equations, their leading coefficients are different from 1 .
From any non-reduced quadratic equation, by dividing both of its parts by the leading coefficient, you can go to the reduced one. This action is an equivalent transformation, that is, the reduced quadratic equation obtained in this way has the same roots as the original non-reduced quadratic equation, or, like it, has no roots.
Let's take an example of how the transition from an unreduced quadratic equation to a reduced one is performed.
Example.
From the equation 3 x 2 +12 x−7=0, go to the corresponding reduced quadratic equation.
Decision.
It is enough for us to perform the division of both parts of the original equation by the leading coefficient 3, it is non-zero, so we can perform this action. We have (3 x 2 +12 x−7):3=0:3 , which is the same as (3 x 2):3+(12 x):3−7:3=0 , and so on (3:3) x 2 +(12:3) x−7:3=0 , whence . So we got the reduced quadratic equation, which is equivalent to the original one.
Answer:
Complete and incomplete quadratic equations
There is a condition a≠0 in the definition of a quadratic equation. This condition is necessary in order for the equation a x 2 +b x+c=0 to be exactly square, since with a=0 it actually becomes a linear equation of the form b x+c=0 .
As for the coefficients b and c, they can be equal to zero, both separately and together. In these cases, the quadratic equation is called incomplete.
Definition.
The quadratic equation a x 2 +b x+c=0 is called incomplete, if at least one of the coefficients b , c is equal to zero.
In its turn
Definition.
Complete quadratic equation is an equation in which all coefficients are different from zero.
These names are not given by chance. This will become clear from the following discussion.
If the coefficient b is equal to zero, then the quadratic equation takes the form a x 2 +0 x+c=0 , and it is equivalent to the equation a x 2 +c=0 . If c=0 , that is, the quadratic equation has the form a x 2 +b x+0=0 , then it can be rewritten as a x 2 +b x=0 . And with b=0 and c=0 we get the quadratic equation a·x 2 =0. The resulting equations differ from the full quadratic equation in that their left-hand sides do not contain either a term with the variable x, or a free term, or both. Hence their name - incomplete quadratic equations.
So the equations x 2 +x+1=0 and −2 x 2 −5 x+0,2=0 are examples of complete quadratic equations, and x 2 =0, −2 x 2 =0, 5 x 2 +3=0 , −x 2 −5 x=0 are incomplete quadratic equations.
Solving incomplete quadratic equations
It follows from the information of the previous paragraph that there is three kinds of incomplete quadratic equations:
- a x 2 =0 , the coefficients b=0 and c=0 correspond to it;
- a x 2 +c=0 when b=0 ;
- and a x 2 +b x=0 when c=0 .
Let us analyze in order how the incomplete quadratic equations of each of these types are solved.
a x 2 \u003d 0
Let's start by solving incomplete quadratic equations in which the coefficients b and c are equal to zero, that is, with equations of the form a x 2 =0. The equation a·x 2 =0 is equivalent to the equation x 2 =0, which is obtained from the original by dividing its both parts by a non-zero number a. Obviously, the root of the equation x 2 \u003d 0 is zero, since 0 2 \u003d 0. This equation has no other roots, which is explained, indeed, for any non-zero number p, the inequality p 2 >0 takes place, which implies that for p≠0, the equality p 2 =0 is never achieved.
So, the incomplete quadratic equation a x 2 \u003d 0 has a single root x \u003d 0.
As an example, we give the solution of an incomplete quadratic equation −4·x 2 =0. It is equivalent to the equation x 2 \u003d 0, its only root is x \u003d 0, therefore, the original equation has a single root zero.
A short solution in this case can be issued as follows:
−4 x 2 \u003d 0,
x 2 \u003d 0,
x=0 .
a x 2 +c=0
Now consider how incomplete quadratic equations are solved, in which the coefficient b is equal to zero, and c≠0, that is, equations of the form a x 2 +c=0. We know that the transfer of a term from one side of the equation to the other with the opposite sign, as well as the division of both sides of the equation by a non-zero number, give an equivalent equation. Therefore, the following equivalent transformations of the incomplete quadratic equation a x 2 +c=0 can be carried out:
- move c to the right side, which gives the equation a x 2 =−c,
- and divide both its parts by a , we get .
The resulting equation allows us to draw conclusions about its roots. Depending on the values of a and c, the value of the expression can be negative (for example, if a=1 and c=2 , then ) or positive, (for example, if a=−2 and c=6 , then ), it is not equal to zero , because by condition c≠0 . We will separately analyze the cases and .
If , then the equation has no roots. This statement follows from the fact that the square of any number is a non-negative number. It follows from this that when , then for any number p the equality cannot be true.
If , then the situation with the roots of the equation is different. In this case, if we recall about, then the root of the equation immediately becomes obvious, it is the number, since. It is easy to guess that the number is also the root of the equation , indeed, . This equation has no other roots, which can be shown, for example, by contradiction. Let's do it.
Let's denote the just voiced roots of the equation as x 1 and −x 1 . Suppose that the equation has another root x 2 different from the indicated roots x 1 and −x 1 . It is known that substitution into the equation instead of x of its roots turns the equation into a true numerical equality. For x 1 and −x 1 we have , and for x 2 we have . The properties of numerical equalities allow us to perform term-by-term subtraction of true numerical equalities, so subtracting the corresponding parts of the equalities gives x 1 2 − x 2 2 =0. The properties of operations with numbers allow us to rewrite the resulting equality as (x 1 − x 2)·(x 1 + x 2)=0 . We know that the product of two numbers is equal to zero if and only if at least one of them is equal to zero. Therefore, it follows from the obtained equality that x 1 −x 2 =0 and/or x 1 +x 2 =0 , which is the same, x 2 =x 1 and/or x 2 = −x 1 . So we have come to a contradiction, since at the beginning we said that the root of the equation x 2 is different from x 1 and −x 1 . This proves that the equation has no other roots than and .
Let's summarize the information in this paragraph. The incomplete quadratic equation a x 2 +c=0 is equivalent to the equation , which
- has no roots if ,
- has two roots and if .
Consider examples of solving incomplete quadratic equations of the form a·x 2 +c=0 .
Let's start with the quadratic equation 9 x 2 +7=0 . After transferring the free term to the right side of the equation, it will take the form 9·x 2 =−7. Dividing both sides of the resulting equation by 9 , we arrive at . Since a negative number is obtained on the right side, this equation has no roots, therefore, the original incomplete quadratic equation 9 x 2 +7=0 has no roots.
Let's solve one more incomplete quadratic equation −x 2 +9=0. We transfer the nine to the right side: -x 2 \u003d -9. Now we divide both parts by −1, we get x 2 =9. The right side contains a positive number, from which we conclude that or . After we write down the final answer: the incomplete quadratic equation −x 2 +9=0 has two roots x=3 or x=−3.
a x 2 +b x=0
It remains to deal with the solution of the last type of incomplete quadratic equations for c=0 . Incomplete quadratic equations of the form a x 2 +b x=0 allows you to solve factorization method. Obviously, we can, located on the left side of the equation, for which it is enough to take the common factor x out of brackets. This allows us to move from the original incomplete quadratic equation to an equivalent equation of the form x·(a·x+b)=0 . And this equation is equivalent to the set of two equations x=0 and a x+b=0 , the last of which is linear and has a root x=−b/a .
So, the incomplete quadratic equation a x 2 +b x=0 has two roots x=0 and x=−b/a.
To consolidate the material, we will analyze the solution of a specific example.
Example.
Solve the equation.
Decision.
We take x out of brackets, this gives the equation. It is equivalent to two equations x=0 and . We solve the resulting linear equation: , and after dividing mixed number on the common fraction, we find . Therefore, the roots of the original equation are x=0 and .
After getting the necessary practice, the solutions of such equations can be written briefly:
Answer:
x=0 , .
Discriminant, formula of the roots of a quadratic equation
To solve quadratic equations, there is a root formula. Let's write down the formula of the roots of the quadratic equation: , where D=b 2 −4 a c- so-called discriminant of a quadratic equation. The notation essentially means that .
It is useful to know how the root formula was obtained, and how it is applied in finding the roots of quadratic equations. Let's deal with this.
Derivation of the formula of the roots of a quadratic equation
Let us need to solve the quadratic equation a·x 2 +b·x+c=0 . Let's perform some equivalent transformations:
- We can divide both parts of this equation by a non-zero number a, as a result we get the reduced quadratic equation.
- Now select a full square on its left side: . After that, the equation will take the form .
- At this stage, it is possible to carry out the transfer of the last two terms to the right side with the opposite sign, we have .
- And let's also transform the expression on the right side: .
As a result, we arrive at the equation , which is equivalent to the original quadratic equation a·x 2 +b·x+c=0 .
We have already solved equations similar in form in the previous paragraphs when we analyzed . This allows us to draw the following conclusions regarding the roots of the equation:
- if , then the equation has no real solutions;
- if , then the equation has the form , therefore, , from which its only root is visible;
- if , then or , which is the same as or , that is, the equation has two roots.
Thus, the presence or absence of the roots of the equation, and hence the original quadratic equation, depends on the sign of the expression on the right side. In turn, the sign of this expression is determined by the sign of the numerator, since the denominator 4 a 2 is always positive, that is, the sign of the expression b 2 −4 a c . This expression b 2 −4 a c is called discriminant of a quadratic equation and marked with the letter D. From here, the essence of the discriminant is clear - by its value and sign, it is concluded whether the quadratic equation has real roots, and if so, what is their number - one or two.
We return to the equation , rewrite it using the notation of the discriminant: . And we conclude:
- if D<0 , то это уравнение не имеет действительных корней;
- if D=0, then this equation has a single root;
- finally, if D>0, then the equation has two roots or , which can be rewritten in the form or , and after expanding and reducing the fractions to a common denominator, we get .
So we derived the formulas for the roots of the quadratic equation, they look like , where the discriminant D is calculated by the formula D=b 2 −4 a c .
With their help, with a positive discriminant, you can calculate both real roots of a quadratic equation. When the discriminant is equal to zero, both formulas give the same root value corresponding to the only solution of the quadratic equation. And with a negative discriminant, when trying to use the formula for the roots of a quadratic equation, we are faced with extracting the square root from a negative number, which takes us beyond and school curriculum. With a negative discriminant, the quadratic equation has no real roots, but has a pair complex conjugate roots, which can be found using the same root formulas we obtained.
Algorithm for solving quadratic equations using root formulas
In practice, when solving a quadratic equation, you can immediately use the root formula, with which to calculate their values. But this is more about finding complex roots.
However, in school course algebra usually we are talking not about complex, but about real roots of a quadratic equation. In this case, it is advisable to first find the discriminant before using the formulas for the roots of the quadratic equation, make sure that it is non-negative (otherwise, we can conclude that the equation has no real roots), and after that calculate the values of the roots.
The above reasoning allows us to write algorithm for solving a quadratic equation. To solve the quadratic equation a x 2 + b x + c \u003d 0, you need:
- using the discriminant formula D=b 2 −4 a c calculate its value;
- conclude that the quadratic equation has no real roots if the discriminant is negative;
- calculate the only root of the equation using the formula if D=0 ;
- find two real roots of a quadratic equation using the root formula if the discriminant is positive.
Here we only note that if the discriminant is equal to zero, the formula can also be used, it will give the same value as .
You can move on to examples of applying the algorithm for solving quadratic equations.
Examples of solving quadratic equations
Consider solutions of three quadratic equations with positive, negative, and zero discriminant. Having dealt with their solution, by analogy it will be possible to solve any other quadratic equation. Let's start.
Example.
Find the roots of the equation x 2 +2 x−6=0 .
Decision.
In this case, we have the following coefficients of the quadratic equation: a=1 , b=2 and c=−6 . According to the algorithm, you first need to calculate the discriminant, for this we substitute the indicated a, b and c into the discriminant formula, we have D=b 2 −4 a c=2 2 −4 1 (−6)=4+24=28. Since 28>0, that is, the discriminant is greater than zero, the quadratic equation has two real roots. Let's find them by the formula of roots , we get , here we can simplify the expressions obtained by doing factoring out the sign of the root followed by fraction reduction:
Answer:
Let's move on to the next typical example.
Example.
Solve the quadratic equation −4 x 2 +28 x−49=0 .
Decision.
We start by finding the discriminant: D=28 2 −4 (−4) (−49)=784−784=0. Therefore, this quadratic equation has a single root, which we find as , that is,
Answer:
x=3.5 .
It remains to consider the solution of quadratic equations with negative discriminant.
Example.
Solve the equation 5 y 2 +6 y+2=0 .
Decision.
Here are the coefficients of the quadratic equation: a=5 , b=6 and c=2 . Substituting these values into the discriminant formula, we have D=b 2 −4 a c=6 2 −4 5 2=36−40=−4. The discriminant is negative, therefore, this quadratic equation has no real roots.
If you need to specify complex roots, then we use the well-known formula for the roots of the quadratic equation, and perform actions with complex numbers
:
Answer:
there are no real roots, the complex roots are: .
Once again, we note that if the discriminant of the quadratic equation is negative, then the school usually immediately writes down the answer, in which they indicate that there are no real roots, and they do not find complex roots.
Root formula for even second coefficients
The formula for the roots of a quadratic equation , where D=b 2 −4 a c allows you to get a more compact formula that allows you to solve quadratic equations with an even coefficient at x (or simply with a coefficient that looks like 2 n, for example, or 14 ln5=2 7 ln5 ). Let's take her out.
Let's say we need to solve a quadratic equation of the form a x 2 +2 n x + c=0 . Let's find its roots using the formula known to us. To do this, we calculate the discriminant D=(2 n) 2 −4 a c=4 n 2 −4 a c=4 (n 2 −a c), and then we use the root formula:
Denote the expression n 2 −a c as D 1 (sometimes it is denoted D "). Then the formula for the roots of the considered quadratic equation with the second coefficient 2 n takes the form 
, where D 1 =n 2 −a c .
It is easy to see that D=4·D 1 , or D 1 =D/4 . In other words, D 1 is the fourth part of the discriminant. It is clear that the sign of D 1 is the same as the sign of D . That is, the sign D 1 is also an indicator of the presence or absence of the roots of the quadratic equation.
So, to solve a quadratic equation with the second coefficient 2 n, you need
- Calculate D 1 =n 2 −a·c ;
- If D 1<0 , то сделать вывод, что действительных корней нет;
- If D 1 =0, then calculate the only root of the equation using the formula;
- If D 1 >0, then find two real roots using the formula.
Consider the solution of the example using the root formula obtained in this paragraph.
Example.
Solve the quadratic equation 5 x 2 −6 x−32=0 .
Decision.
The second coefficient of this equation can be represented as 2·(−3) . That is, you can rewrite the original quadratic equation in the form 5 x 2 +2 (−3) x−32=0 , here a=5 , n=−3 and c=−32 , and calculate the fourth part of the discriminant: D 1 =n 2 −a c=(−3) 2 −5 (−32)=9+160=169. Since its value is positive, the equation has two real roots. We find them using the corresponding root formula:
Note that it was possible to use the usual formula for the roots of a quadratic equation, but in this case, more computational work would have to be done.
Answer:
Simplification of the form of quadratic equations
Sometimes, before embarking on the calculation of the roots of a quadratic equation using formulas, it does not hurt to ask the question: “Is it possible to simplify the form of this equation”? Agree that in terms of calculations it will be easier to solve the quadratic equation 11 x 2 −4 x −6=0 than 1100 x 2 −400 x−600=0 .
Usually, a simplification of the form of a quadratic equation is achieved by multiplying or dividing both sides of it by some number. For example, in the previous paragraph, we managed to achieve a simplification of the equation 1100 x 2 −400 x −600=0 by dividing both sides by 100 .
A similar transformation is carried out with quadratic equations, the coefficients of which are not . In this case, both parts of the equation are usually divided by the absolute values of its coefficients. For example, let's take the quadratic equation 12 x 2 −42 x+48=0. absolute values of its coefficients: gcd(12, 42, 48)= gcd(gcd(12, 42), 48)= gcd(6, 48)=6 . Dividing both parts of the original quadratic equation by 6 , we arrive at the equivalent quadratic equation 2 x 2 −7 x+8=0 .
And the multiplication of both parts of the quadratic equation is usually done to get rid of fractional coefficients. In this case, the multiplication is carried out on the denominators of its coefficients. For example, if both parts of a quadratic equation are multiplied by LCM(6, 3, 1)=6 , then it will take a simpler form x 2 +4 x−18=0 .
In conclusion of this paragraph, we note that almost always get rid of the minus at the highest coefficient of the quadratic equation by changing the signs of all terms, which corresponds to multiplying (or dividing) both parts by −1. For example, usually from the quadratic equation −2·x 2 −3·x+7=0 go to the solution 2·x 2 +3·x−7=0 .
Relationship between roots and coefficients of a quadratic equation
The formula for the roots of a quadratic equation expresses the roots of an equation in terms of its coefficients. Based on the formula of the roots, you can get other relationships between the roots and coefficients.
The most well-known and applicable formulas from the Vieta theorem of the form and . In particular, for the given quadratic equation, the sum of the roots is equal to the second coefficient with the opposite sign, and the product of the roots is the free term. For example, by the form of the quadratic equation 3 x 2 −7 x+22=0, we can immediately say that the sum of its roots is 7/3, and the product of the roots is 22/3.
Using the already written formulas, you can get a number of other relationships between the roots and coefficients of the quadratic equation. For example, you can express the sum of the squares of the roots of a quadratic equation in terms of its coefficients: .
Bibliography.
- Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Mordkovich A. G. Algebra. 8th grade. At 2 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemozina, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.
”, that is, equations of the first degree. In this lesson, we will explore what is a quadratic equation and how to solve it.
What is a quadratic equation
Important!
The degree of an equation is determined by the highest degree to which the unknown stands.
If the maximum degree to which the unknown stands is “2”, then you have a quadratic equation.
Examples of quadratic equations
- 5x2 - 14x + 17 = 0
- −x 2 + x +
= 01 3 - x2 + 0.25x = 0
- x 2 − 8 = 0
Important! The general form of the quadratic equation looks like this:
A x 2 + b x + c = 0
"a", "b" and "c" - given numbers.- "a" - the first or senior coefficient;
- "b" - the second coefficient;
- "c" is a free member.
To find "a", "b" and "c" You need to compare your equation with the general form of the quadratic equation "ax 2 + bx + c \u003d 0".
Let's practice determining the coefficients "a", "b" and "c" in quadratic equations.
| The equation | Odds | |||
|---|---|---|---|---|
|
||||
|
||||
| 1 |
| 3 |
- a = −1
- b = 1
- c =
1 3
- a = 1
- b = 0.25
- c = 0
- a = 1
- b = 0
- c = −8
How to solve quadratic equations
Unlike linear equations to solve quadratic equations, a special formula for finding roots.
Remember!
To solve a quadratic equation you need:
- bring the quadratic equation to the general form "ax 2 + bx + c \u003d 0". That is, only "0" should remain on the right side;
- use the formula for roots:
Let's use an example to figure out how to apply the formula to find the roots of a quadratic equation. Let's solve the quadratic equation.
X 2 - 3x - 4 = 0
The equation "x 2 - 3x - 4 = 0" has already been reduced to the general form "ax 2 + bx + c = 0" and does not require additional simplifications. To solve it, we need only apply formula for finding the roots of a quadratic equation.
Let's define the coefficients "a", "b" and "c" for this equation.
x 1;2 =
x 1;2 =
x 1;2 =
x 1;2 =
With its help, any quadratic equation is solved.
In the formula "x 1; 2 \u003d" the root expression is often replaced
"b 2 − 4ac" to the letter "D" and called discriminant. The concept of a discriminant is discussed in more detail in the lesson "What is a discriminant".
Consider another example of a quadratic equation.
x 2 + 9 + x = 7x
In this form, it is rather difficult to determine the coefficients "a", "b", and "c". Let's first bring the equation to the general form "ax 2 + bx + c \u003d 0".
X 2 + 9 + x = 7x
x 2 + 9 + x − 7x = 0
x2 + 9 - 6x = 0
x 2 − 6x + 9 = 0
Now you can use the formula for the roots.
X 1;2 =
x 1;2 =
x 1;2 =
x 1;2 =
x=
| 6 |
| 2 |
x=3
Answer: x = 3
There are times when there are no roots in quadratic equations. This situation occurs when a negative number appears in the formula under the root.
We remind you that the complete quadratic equation is an equation of the form:
Solving full quadratic equations is a bit more complicated (just a little bit) than those given.
Remember, any quadratic equation can be solved using the discriminant!
Even incomplete.
The rest of the methods will help you do it faster, but if you have problems with quadratic equations, first master the solution using the discriminant.
1. Solving quadratic equations using the discriminant.
Solving quadratic equations in this way is very simple, the main thing is to remember the sequence of actions and a couple of formulas.
If, then the equation has 2 roots. Need Special attention turn to step 2.
The discriminant D tells us the number of roots of the equation.
- If, then the formula at the step will be reduced to. Thus, the equation will have only a root.
- If, then we will not be able to extract the root of the discriminant at the step. This indicates that the equation has no roots.
Let's turn to geometric sense quadratic equation.
The graph of the function is a parabola:
Let's go back to our equations and look at a few examples.
Example 9
Solve the Equation
Step 1 skip.
Step 2
Finding the discriminant:
So the equation has two roots.
Step 3
Answer:
Example 10
Solve the Equation
The equation is in standard form, so Step 1 skip.
Step 2
Finding the discriminant:
So the equation has one root.
Answer:
Example 11
Solve the Equation
The equation is in standard form, so Step 1 skip.
Step 2
Finding the discriminant:
This means that we will not be able to extract the root from the discriminant. There are no roots of the equation.
Now we know how to write down such answers correctly.
Answer: no roots
2. Solving quadratic equations using the Vieta theorem
If you remember, then there is such a type of equations that are called reduced (when the coefficient a is equal to):
Such equations are very easy to solve using Vieta's theorem:
The sum of the roots given quadratic equation is equal, and the product of the roots is equal.
You just need to choose a pair of numbers whose product is equal to the free term of the equation, and the sum is equal to the second coefficient, taken with the opposite sign.
Example 12
Solve the Equation
This equation is suitable for solution using Vieta's theorem, because .
The sum of the roots of the equation is, i.e. we get the first equation:
And the product is:
Let's create and solve the system:
- and. The sum is;
- and. The sum is;
- and. The amount is equal.
and are the solution of the system:
Answer: ; .
Example 13
Solve the Equation
Answer:
Example 14
Solve the Equation
The equation is reduced, which means:
Answer:
QUADRATIC EQUATIONS. MIDDLE LEVEL
What is a quadratic equation?
In other words, a quadratic equation is an equation of the form, where - unknown, - some numbers, moreover.
The number is called the highest or first coefficient quadratic equation, - second coefficient, a - free member.
Because if, the equation will immediately become linear, because will disappear.
In this case, and can be equal to zero. In this chair equation is called incomplete.
If all the terms are in place, that is, the equation - complete.
Methods for solving incomplete quadratic equations
To begin with, we will analyze the methods for solving incomplete quadratic equations - they are simpler.
The following types of equations can be distinguished:
I. , in this equation the coefficient and the free term are equal.
II. , in this equation the coefficient is equal.
III. , in this equation the free term is equal to.
Now consider the solution of each of these subtypes.
Obviously, this equation always has only one root:
A number squared cannot be negative, because when multiplying two negative or two positive numbers, the result will always be a positive number. So:
if, then the equation has no solutions;
if we have two roots
These formulas do not need to be memorized. The main thing to remember is that it cannot be less.
Examples of solving quadratic equations
Example 15
Answer:
Never forget about roots with a negative sign!
Example 16
The square of a number cannot be negative, which means that the equation
no roots.
To briefly write that the problem has no solutions, we use the empty set icon.
Answer:
Example 17
So, this equation has two roots: and.
Answer:
Let's take the common factor out of brackets:
The product is equal to zero if at least one of the factors is equal to zero. This means that the equation has a solution when:
So, this quadratic equation has two roots: and.
Example:
Solve the equation.
Decision:
We factorize the left side of the equation and find the roots:
Answer:
Methods for solving complete quadratic equations
1. Discriminant
Solving quadratic equations in this way is easy, the main thing is to remember the sequence of actions and a couple of formulas. Remember, any quadratic equation can be solved using the discriminant! Even incomplete.
Did you notice the root of the discriminant in the root formula?
But the discriminant can be negative.
What to do?
We need to pay special attention to step 2. The discriminant tells us the number of roots of the equation.
- If, then the equation has a root:
- If, then the equation has the same root, but in fact, one root:
Such roots are called double roots.
- If, then the root of the discriminant is not extracted. This indicates that the equation has no roots.
Why are there different numbers of roots?
Let us turn to the geometric meaning of the quadratic equation. The graph of the function is a parabola:
In a particular case, which is a quadratic equation, .
And this means that the roots of the quadratic equation are the points of intersection with the x-axis (axis).
The parabola may not cross the axis at all, or it may intersect it at one (when the top of the parabola lies on the axis) or two points.
In addition, the coefficient is responsible for the direction of the branches of the parabola. If, then the branches of the parabola are directed upwards, and if - then downwards.
4 examples of solving quadratic equations
Example 18
Answer:
Example 19
Answer: .
Example 20
Answer:
Example 21
This means there are no solutions.
Answer: .
2. Vieta's theorem
Using Vieta's theorem is very easy.
All you need is pick up such a pair of numbers, the product of which is equal to the free term of the equation, and the sum is equal to the second coefficient, taken with the opposite sign.
It is important to remember that Vieta's theorem can only be applied to given quadratic equations ().
Let's look at a few examples:
Example 22
Solve the equation.
Decision:
This equation is suitable for solution using Vieta's theorem, because . Other coefficients: ; .
The sum of the roots of the equation is:
And the product is:
Let's select such pairs of numbers, the product of which is equal, and check if their sum is equal:
- and. The sum is;
- and. The sum is;
- and. The amount is equal.
and are the solution of the system:
Thus, and are the roots of our equation.
Answer: ; .
Example 23
Decision:
We select such pairs of numbers that give in the product, and then check whether their sum is equal:
and: give in total.
and: give in total. To get it, you just need to change the signs of the alleged roots: and, after all, the work.
Answer:
Example 24
Decision:
The free term of the equation is negative, and hence the product of the roots is a negative number. This is possible only if one of the roots is negative and the other is positive. So the sum of the roots is differences of their modules.
We select such pairs of numbers that give in the product, and the difference of which is equal to:
and: their difference is - not suitable;
and: - not suitable;
and: - not suitable;
and: - suitable. It remains only to remember that one of the roots is negative. Since their sum must be equal, then the root, which is smaller in absolute value, must be negative: . We check:
Answer:
Example 25
Solve the equation.
Decision:
The equation is reduced, which means:
The free term is negative, and hence the product of the roots is negative. And this is possible only when one root of the equation is negative and the other is positive.
We select such pairs of numbers whose product is equal, and then determine which roots should have a negative sign:
Obviously, only roots and are suitable for the first condition:
Answer:
Example 26
Solve the equation.
Decision:
The equation is reduced, which means:
The sum of the roots is negative, which means that at least one of the roots is negative. But since their product is positive, it means both roots are minus.
We select such pairs of numbers, the product of which is equal to:
Obviously, the roots are the numbers and.
Answer:
Agree, it is very convenient - to invent roots orally, instead of counting this nasty discriminant.
Try to use Vieta's theorem as often as possible!
But the Vieta theorem is needed in order to facilitate and speed up finding the roots.
To make it profitable for you to use it, you must bring the actions to automatism. And for this, solve five more examples.
But don't cheat: you can't use the discriminant! Only Vieta's theorem!
5 examples of Vieta's theorem for self-study
Example 27
Task 1. ((x)^(2))-8x+12=0
According to Vieta's theorem:
As usual, we start the selection with the product:
Not suitable because the amount;
: the amount is what you need.
Answer: ; .
Example 28
Task 2.
And again, our favorite Vieta theorem: the sum should work out, but the product is equal.
But since it should be not, but, we change the signs of the roots: and (in total).
Answer: ; .
Example 29
Task 3.
Hmm... Where is it?
It is necessary to transfer all the terms into one part:
The sum of the roots is equal to the product.
Yes, stop! The equation is not given.
But Vieta's theorem is applicable only in the given equations.
So first you need to bring the equation.
If you can’t bring it up, drop this idea and solve it in another way (for example, through the discriminant).
Let me remind you that to bring a quadratic equation means to make the leading coefficient equal to:
Then the sum of the roots is equal, and the product.
It's easier to pick up here: after all - a prime number (sorry for the tautology).
Answer: ; .
Example 30
Task 4.
The free term is negative.
What's so special about it?
And the fact that the roots will be of different signs.
And now, during the selection, we check not the sum of the roots, but the difference between their modules: this difference is equal, but the product.
So, the roots are equal and, but one of them is with a minus.
Vieta's theorem tells us that the sum of the roots is equal to the second coefficient with the opposite sign, that is.
This means that the smaller root will have a minus: and, since.
Answer: ; .
Example 31
Task 5.
What needs to be done first?
That's right, give the equation:
Again: we select the factors of the number, and their difference should be equal to:
The roots are equal and, but one of them is minus. Which? Their sum must be equal, which means that with a minus there will be a larger root.
Answer: ; .
Summarize
- Vieta's theorem is used only in the given quadratic equations.
- Using the Vieta theorem, you can find the roots by selection, orally.
- If the equation is not given or no suitable pair of factors of the free term was found, then there are no integer roots, and you need to solve it in another way (for example, through the discriminant).
3. Full square selection method
If all the terms containing the unknown are represented as terms from the formulas of abbreviated multiplication - the square of the sum or difference - then after the change of variables, the equation can be represented as an incomplete quadratic equation of the type.
For example:
Example 32
Solve the equation: .
Decision:
Answer:
Example 33
Solve the equation: .
Decision:
Answer:
In general, the transformation will look like this:
This implies: .
Doesn't it remind you of anything?
It's the discriminant! That's exactly how the discriminant formula was obtained.
QUADRATIC EQUATIONS. BRIEFLY ABOUT THE MAIN
Quadratic equation is an equation of the form, where is the unknown, are the coefficients of the quadratic equation, is the free term.
Complete quadratic equation- an equation in which the coefficients are not equal to zero.
Reduced quadratic equation- an equation in which the coefficient, that is: .
Incomplete quadratic equation- an equation in which the coefficient and or free term c are equal to zero:
- if the coefficient, the equation has the form: ,
- if a free term, the equation has the form: ,
- if and, the equation has the form: .
1. Algorithm for solving incomplete quadratic equations
1.1. An incomplete quadratic equation of the form, where, :
1) Express the unknown: ,
2) Check the sign of the expression:
- if, then the equation has no solutions,
- if, then the equation has two roots.
1.2. An incomplete quadratic equation of the form, where, :
1) Let's take the common factor out of brackets: ,
2) The product is equal to zero if at least one of the factors is equal to zero. Therefore, the equation has two roots:
1.3. An incomplete quadratic equation of the form, where:
This equation always has only one root: .
2. Algorithm for solving complete quadratic equations of the form where
2.1. Solution using the discriminant
1) Let's bring the equation to the standard form: ,
2) Calculate the discriminant using the formula: , which indicates the number of roots of the equation:
3) Find the roots of the equation:
- if, then the equation has a root, which are found by the formula:
- if, then the equation has a root, which is found by the formula:
- if, then the equation has no roots.
2.2. Solution using Vieta's theorem
The sum of the roots of the reduced quadratic equation (an equation of the form, where) is equal, and the product of the roots is equal, i.e. , a.
2.3. Full square solution
Quadratic equations are studied in grade 8, so there is nothing complicated here. The ability to solve them is essential.
A quadratic equation is an equation of the form ax 2 + bx + c = 0, where the coefficients a , b and c are arbitrary numbers, and a ≠ 0.
Before studying specific methods of solving, we note that all quadratic equations can be divided into three classes:
- Have no roots;
- They have exactly one root;
- They have two different roots.
This is an important difference between quadratic and linear equations, where the root always exists and is unique. How to determine how many roots an equation has? There is a wonderful thing for this - discriminant.
Discriminant
Let the quadratic equation ax 2 + bx + c = 0 be given. Then the discriminant is simply the number D = b 2 − 4ac .
This formula must be known by heart. Where it comes from is not important now. Another thing is important: by the sign of the discriminant, you can determine how many roots a quadratic equation has. Namely:
- If D< 0, корней нет;
- If D = 0, there is exactly one root;
- If D > 0, there will be two roots.
Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many people think. Take a look at the examples and you will understand everything yourself:
Task. How many roots do quadratic equations have:
- x 2 - 8x + 12 = 0;
- 5x2 + 3x + 7 = 0;
- x 2 − 6x + 9 = 0.
We write the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 − 4 1 12 = 64 − 48 = 16
So, the discriminant is positive, so the equation has two different roots. We analyze the second equation in the same way:
a = 5; b = 3; c = 7;
D \u003d 3 2 - 4 5 7 \u003d 9 - 140 \u003d -131.
The discriminant is negative, there are no roots. The last equation remains:
a = 1; b = -6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.
The discriminant is equal to zero - the root will be one.
Note that coefficients have been written out for each equation. Yes, it's long, yes, it's tedious - but you won't mix up the odds and don't make stupid mistakes. Choose for yourself: speed or quality.
By the way, if you “fill your hand”, after a while you will no longer need to write out all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not so many.
The roots of a quadratic equation
Now let's move on to the solution. If the discriminant D > 0, the roots can be found using the formulas:
The basic formula for the roots of a quadratic equation
When D = 0, you can use any of these formulas - you get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.
- x 2 - 2x - 3 = 0;
- 15 - 2x - x2 = 0;
- x2 + 12x + 36 = 0.
First equation:
x 2 - 2x - 3 = 0 ⇒ a = 1; b = −2; c = -3;
D = (−2) 2 − 4 1 (−3) = 16.
D > 0 ⇒ the equation has two roots. Let's find them:
Second equation:
15 − 2x − x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 − 4 (−1) 15 = 64.
D > 0 ⇒ the equation again has two roots. Let's find them
\[\begin(align) & ((x)_(1))=\frac(2+\sqrt(64))(2\cdot \left(-1 \right))=-5; \\ & ((x)_(2))=\frac(2-\sqrt(64))(2\cdot \left(-1 \right))=3. \\ \end(align)\]
Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 − 4 1 36 = 0.
D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:
As you can see from the examples, everything is very simple. If you know the formulas and be able to count, there will be no problems. Most often, errors occur when negative coefficients are substituted into the formula. Here, again, the technique described above will help: look at the formula literally, paint each step - and get rid of mistakes very soon.
Incomplete quadratic equations
It happens that the quadratic equation is somewhat different from what is given in the definition. For example:
- x2 + 9x = 0;
- x2 − 16 = 0.
It is easy to see that one of the terms is missing in these equations. Such quadratic equations are even easier to solve than standard ones: they do not even need to calculate the discriminant. So let's introduce a new concept:
The equation ax 2 + bx + c = 0 is called an incomplete quadratic equation if b = 0 or c = 0, i.e. the coefficient of the variable x or the free element is equal to zero.
Of course, a very difficult case is possible when both of these coefficients are equal to zero: b \u003d c \u003d 0. In this case, the equation takes the form ax 2 \u003d 0. Obviously, such an equation has a single root: x \u003d 0.
Let's consider other cases. Let b \u003d 0, then we get an incomplete quadratic equation of the form ax 2 + c \u003d 0. Let's slightly transform it:
Because arithmetic Square root exists only from a non-negative number, the last equality makes sense only for (−c /a ) ≥ 0. Conclusion:
- If an incomplete quadratic equation of the form ax 2 + c = 0 satisfies the inequality (−c / a ) ≥ 0, there will be two roots. The formula is given above;
- If (−c / a )< 0, корней нет.
As you can see, the discriminant was not required - there are no complex calculations at all in incomplete quadratic equations. In fact, it is not even necessary to remember the inequality (−c / a ) ≥ 0. It is enough to express the value of x 2 and see what is on the other side of the equal sign. If there is a positive number, there will be two roots. If negative, there will be no roots at all.
Now let's deal with equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factorize the polynomial:
Taking the common factor out of the bracketThe product is equal to zero when at least one of the factors is equal to zero. This is where the roots come from. In conclusion, we will analyze several of these equations:
Task. Solve quadratic equations:
- x2 − 7x = 0;
- 5x2 + 30 = 0;
- 4x2 − 9 = 0.
x 2 − 7x = 0 ⇒ x (x − 7) = 0 ⇒ x 1 = 0; x2 = −(−7)/1 = 7.
5x2 + 30 = 0 ⇒ 5x2 = -30 ⇒ x2 = -6. There are no roots, because the square cannot be equal to a negative number.
4x 2 − 9 = 0 ⇒ 4x 2 = 9 ⇒ x 2 = 9/4 ⇒ x 1 = 3/2 = 1.5; x 2 \u003d -1.5.
Formulas for the roots of a quadratic equation. The cases of real, multiple and complex roots are considered. Factorization of a square trinomial. Geometric interpretation. Examples of determining roots and factorization.
ContentSee also: Solving quadratic equations online
Basic Formulas
Consider the quadratic equation:
(1)
.
The roots of a quadratic equation(1) are determined by the formulas:
;
.
These formulas can be combined like this:
.
When the roots of the quadratic equation are known, then the polynomial of the second degree can be represented as a product of factors (factored):
.
Further, we consider that - real numbers.
Consider discriminant of a quadratic equation:
.
If the discriminant is positive, then the quadratic equation (1) has two different real roots:
;
.
Then the factorization of the square trinomial has the form:
.
If the discriminant is zero, then the quadratic equation (1) has two multiple (equal) real roots:
.
Factorization:
.
If the discriminant is negative, then the quadratic equation (1) has two complex conjugate roots:
;
.
Here is the imaginary unit, ;
and are the real and imaginary parts of the roots:
;
.
Then
.
Graphic interpretation
If build function graph
,
which is a parabola, then the points of intersection of the graph with the axis will be the roots of the equation
.
When , the graph crosses the abscissa axis (axis) at two points ().
When , the graph touches the x-axis at one point ().
When , the graph does not cross the x-axis ().
Useful Formulas Related to Quadratic Equation
(f.1) ;
(f.2) ;
(f.3) .
Derivation of the formula for the roots of a quadratic equation
We perform transformations and apply formulas (f.1) and (f.3):
,
where
;
.
So, we got the formula for the polynomial of the second degree in the form:
.
From this it can be seen that the equation
performed at
and .
That is, and are the roots of the quadratic equation
.
Examples of determining the roots of a quadratic equation
Example 1
(1.1)
.
.
Comparing with our equation (1.1), we find the values of the coefficients:
.
Finding the discriminant:
.
Since the discriminant is positive, the equation has two real roots:
;
;
.
From here we obtain the decomposition of the square trinomial into factors:
.
Graph of the function y = 2 x 2 + 7 x + 3 crosses the x-axis at two points.
Let's plot the function
.
The graph of this function is a parabola. It crosses the x-axis (axis) at two points:
and .
These points are the roots of the original equation (1.1).
;
;
.
Example 2
Find the roots of a quadratic equation:
(2.1)
.
We write the quadratic equation in general form:
.
Comparing with the original equation (2.1), we find the values of the coefficients:
.
Finding the discriminant:
.
Since the discriminant is zero, the equation has two multiple (equal) roots:
;
.
Then the factorization of the trinomial has the form:
.
Graph of the function y = x 2 - 4 x + 4 touches the x-axis at one point.
Let's plot the function
.
The graph of this function is a parabola. It touches the x-axis (axis) at one point:
.
This point is the root of the original equation (2.1). Since this root is factored twice:
,
then such a root is called a multiple. That is, they consider that there are two equal roots:
.
;
.
Example 3
Find the roots of a quadratic equation:
(3.1)
.
We write the quadratic equation in general form:
(1)
.
Let us rewrite the original equation (3.1):
.
Comparing with (1), we find the values of the coefficients:
.
Finding the discriminant:
.
The discriminant is negative, . Therefore, there are no real roots.
You can find complex roots:
;
;
.
Then
.
The graph of the function does not cross the x-axis. There are no real roots.
Let's plot the function
.
The graph of this function is a parabola. It does not cross the abscissa (axis). Therefore, there are no real roots.
There are no real roots. Complex roots:
;
;
.