We continue to study the topic “ solving equations" We have already become acquainted with linear equations and are moving on to getting acquainted with quadratic equations.
First, we will look at what a quadratic equation is, how it is written in general form, and give related definitions. After this, we will use examples to examine in detail how incomplete quadratic equations are solved. Let's move on to the solution complete equations, we get the root formula, get acquainted with the discriminant quadratic equation and consider solutions to typical examples. Finally, let's trace the connections between the 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 a conversation about quadratic equations with the definition of a quadratic equation, as well as related definitions. After this, you can consider the main types of quadratic equations: reduced and unreduced, 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 non-zero.
Let's say right away that quadratic equations are often called equations of the second degree. This is due to the fact that the quadratic equation is algebraic equation second degree.
The stated 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. These are quadratic equations.
Definition.
Numbers a, b and c are called coefficients of the quadratic equation a·x 2 +b·x+c=0, and coefficient a is called the first, or the highest, or the coefficient of x 2, b is the second coefficient, or the coefficient of x, and c is the free term.
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 free member is equal to −3. Please note that when the coefficients b and/or c are negative, as in the example just given, the short form of the quadratic equation is 5 x 2 −2 x−3=0 , rather than 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 quadratic equation, which is due to the peculiarities of writing such. For example, in the quadratic equation y 2 −y+3=0 the leading coefficient is one, and the coefficient of y is equal to −1.
Reduced and unreduced quadratic equations
Depending on the value of the leading coefficient, reduced and unreduced quadratic equations are distinguished. Let us give the corresponding definitions.
Definition.
A quadratic equation in which the leading coefficient is 1 is called given quadratic equation. Otherwise the quadratic equation is untouched.
According to this definition, quadratic equations x 2 −3·x+1=0, x 2 −x−2/3=0, etc. – given, in each of them the first coefficient is equal to one. A 5 x 2 −x−1=0, etc. - unreduced quadratic equations, their leading coefficients are different from 1.
From any unreduced quadratic equation, by dividing both sides 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 unreduced quadratic equation, or, like it, has no roots.
Let us look at 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.
Solution.
We just need to divide both sides 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, (3 x 2):3+(12 x):3−7:3=0, and then (3:3) x 2 +(12:3) x−7:3=0, from where . This is how we obtained the reduced quadratic equation, which is equivalent to the original one.
Answer:
Complete and incomplete quadratic equations
The definition of a quadratic equation contains the condition a≠0. This condition is necessary so that the equation a x 2 + b x + c = 0 is quadratic, since when 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 individually 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 turn
Definition.
Complete quadratic equation is an equation in which all coefficients are different from zero.
Such names were not given by chance. This will become clear from the following discussions.
If the coefficient b is 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 complete 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
From the information in the previous paragraph it follows that there is three types 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 examine in order how incomplete quadratic equations of each of these types are solved.
a x 2 =0
Let's start with 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 both parts by a non-zero number a. Obviously, the root of the equation x 2 =0 is zero, since 0 2 =0. This equation has no other roots, which is explained by the fact that for any non-zero number p the inequality p 2 >0 holds, which means that for p≠0 the equality p 2 =0 is never achieved.
So, the incomplete quadratic equation a·x 2 =0 has a single root x=0.
As an example, we give the solution to the incomplete quadratic equation −4 x 2 =0. It is equivalent to the equation x 2 =0, its only root is x=0, therefore, the original equation has a single root zero.
A short solution in this case can be written as follows:
−4 x 2 =0 ,
x 2 =0,
x=0 .
a x 2 +c=0
Now let's look at how incomplete quadratic equations are solved in which the coefficient b is zero and c≠0, that is, equations of the form a x 2 +c=0. We know that moving a term from one side of the equation to the other with the opposite sign, as well as dividing both sides of the equation by a non-zero number, gives an equivalent equation. Therefore, we can carry out the following equivalent transformations incomplete quadratic equation a x 2 +c=0 :
- move c to the right side, which gives the equation a x 2 =−c,
- and divide both sides 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 , since by condition c≠0. Let's look at the cases separately.
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 remember about , then the root of the equation immediately becomes obvious; it is the number, since . It’s 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 this.
Let us denote the roots of the equation just announced as x 1 and −x 1 . Suppose that the equation has one more root x 2, different from the indicated roots x 1 and −x 1. It is known that substituting its roots into an equation instead of x turns the equation into a correct 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 correct 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, from the resulting equality it follows 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 came 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 roots other than and .
Let us summarize the information in this paragraph. The incomplete quadratic equation a x 2 +c=0 is equivalent to the equation that
- has no roots if ,
- has two roots and , if .
Let's 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 moving 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 on the right side it turned out negative number, then this equation has no roots, therefore, the original incomplete quadratic equation 9 x 2 +7=0 has no roots.
Let's solve another incomplete quadratic equation −x 2 +9=0. We move the nine to the right side: −x 2 =−9. Now we divide both sides by −1, we get x 2 =9. On the right side there is a positive number, from which we conclude that or . Then 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 equivalent equation of the form x·(a·x+b)=0 . And this equation is equivalent to a set of two equations x=0 and a·x+b=0, the latter 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 to a specific example.
Example.
Solve the equation.
Solution.
Taking x out of brackets gives the equation . It is equivalent to two equations x=0 and . We solve the resulting linear equation: , and perform the division mixed number on common fraction, we find . Therefore, the roots of the original equation are x=0 and .
After gaining the necessary practice, solutions to such equations can be written briefly:
Answer:
x=0 , .
Discriminant, formula for the roots of a quadratic equation
To solve quadratic equations, there is a root formula. Let's write it down formula for the roots of a quadratic equation: , Where D=b 2 −4 a c- so-called discriminant of a quadratic equation. The entry essentially means that .
It is useful to know how the root formula was derived and how it is used in finding the roots of quadratic equations. Let's figure this out.
Derivation of the formula for 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 sides of this equation by a non-zero number a, resulting in the following quadratic equation.
- Now let's highlight perfect square on its left side: . After this, the equation will take the form .
- At this stage, it is possible to transfer 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 an equation that 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 examined. 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 roots of the equation, and therefore 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, by the sign of the expression b 2 −4·a·c. This expression b 2 −4 a c was called discriminant of a quadratic equation and designated by the letter D. From here the essence of the discriminant is clear - based on its value and sign, they conclude whether the quadratic equation has real roots, and if so, what is their number - one or two.
Let's return to the equation and rewrite it using the discriminant notation: . And we draw conclusions:
- 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 bringing the fractions to a common denominator we obtain.
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 value of the root, corresponding to the only solution 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 of a negative number, which takes us beyond the scope 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 quadratic equations, you can immediately use the root formula to calculate their values. But this is more related to finding complex roots.
However, in school course Algebra usually deals not with complex, but with real roots of a quadratic equation. In this case, it is advisable, before using the formulas for the roots of a quadratic equation, to first find the discriminant, make sure that it is non-negative (otherwise, we can conclude that the equation does not have real roots), and only then 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=0, you need to:
- using the discriminant formula D=b 2 −4·a·c, calculate its value;
- conclude that a 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 just note that if the discriminant is equal to zero, you can also use the formula; it will give the same value as .
You can move on to examples of using the algorithm for solving quadratic equations.
Examples of solving quadratic equations
Let's consider solutions to three quadratic equations with a positive, negative and zero discriminant. Having dealt with their solution, by analogy it will be possible to solve any other quadratic equation. Let's begin.
Example.
Find the roots of the equation x 2 +2·x−6=0.
Solution.
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; to do 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 using the root formula, we get, here you can simplify the resulting expressions by doing moving the multiplier beyond the root sign followed by reduction of the fraction:
Answer:
Let's move on to the next typical example.
Example.
Solve the quadratic equation −4 x 2 +28 x−49=0 .
Solution.
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 solving quadratic equations with a negative discriminant.
Example.
Solve the equation 5·y 2 +6·y+2=0.
Solution.
Here are the coefficients of the quadratic equation: a=5, b=6 and c=2. We substitute 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 indicate complex roots, then we apply the well-known formula for the roots of a quadratic equation, and perform actions with complex numbers
:
Answer:
there are no real roots, complex roots are: .
Let us note once again that if the discriminant of a quadratic equation is negative, then in school they usually immediately write down an answer in which they indicate that there are no real roots, and complex roots are not found.
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 obtain a formula of a more compact form, allowing you to solve quadratic equations with an even coefficient for x (or simply with a coefficient of the form 2·n, for example, or 14· ln5=2·7·ln5 ). Let's get 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 we know. 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:
Let us denote the expression n 2 −a c as D 1 (sometimes it is denoted D "). Then the formula for the roots of the quadratic equation under consideration with the second coefficient 2 n will take 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 roots of a quadratic equation.
So, to solve a quadratic equation with a 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.
Let's consider solving the example using the root formula obtained in this paragraph.
Example.
Solve the quadratic equation 5 x 2 −6 x −32=0 .
Solution.
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. Let's find them using the appropriate 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 performed.
Answer:
Simplifying the form of quadratic equations
Sometimes, before starting to calculate the roots of a quadratic equation using formulas, it doesn’t 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.
Typically, simplifying the form of a quadratic equation is achieved by multiplying or dividing both sides by a certain number. For example, in the previous paragraph it was possible to simplify 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, we usually divide both sides of the equation by absolute values 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 sides of the original quadratic equation by 6, we arrive at the equivalent quadratic equation 2 x 2 −7 x+8=0.
And multiplying both sides of a quadratic equation is usually done to get rid of fractional coefficients. In this case, multiplication is carried out by the denominators of its coefficients. For example, if both sides of the quadratic equation are multiplied by LCM(6, 3, 1)=6, then it will take the simpler form x 2 +4·x−18=0.
In conclusion of this point, we note that they almost always get rid of the minus at the highest coefficient of a quadratic equation by changing the signs of all terms, which corresponds to multiplying (or dividing) both sides by −1. For example, usually one moves from the quadratic equation −2 x 2 −3 x+7=0 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 the equation through its coefficients. Based on the root formula, you can obtain other relationships between roots and coefficients.
The most well-known and applicable formulas from Vieta’s theorem are 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 equal to 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 equal to 7/3, and the product of the roots is equal to 22/3.
Using the already written formulas, you can obtain a number of other connections 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 through its coefficients: .
References.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by 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 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich. - 11th ed., erased. - M.: Mnemosyne, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.
I hope that after studying this article you will learn how to find the roots of a complete quadratic equation.
Using the discriminant, only complete quadratic equations are solved; to solve incomplete quadratic equations, other methods are used, which you will find in the article “Solving incomplete quadratic equations.”
What quadratic equations are called complete? This equations of the form ax 2 + b x + c = 0, where coefficients a, b and c are not equal to zero. So, to solve a complete quadratic equation, we need to calculate the discriminant D.
D = b 2 – 4ac.
Depending on the value of the discriminant, we will write down the answer.
If the discriminant is a negative number (D< 0),то корней нет.
If the discriminant is zero, then x = (-b)/2a. When the discriminant is a positive number (D > 0),
then x 1 = (-b - √D)/2a, and x 2 = (-b + √D)/2a.
For example. Solve the equation x 2– 4x + 4= 0.
D = 4 2 – 4 4 = 0
x = (- (-4))/2 = 2
Answer: 2.
Solve Equation 2 x 2 + x + 3 = 0.
D = 1 2 – 4 2 3 = – 23
Answer: no roots.
Solve Equation 2 x 2 + 5x – 7 = 0.
D = 5 2 – 4 2 (–7) = 81
x 1 = (-5 - √81)/(2 2)= (-5 - 9)/4= – 3.5
x 2 = (-5 + √81)/(2 2) = (-5 + 9)/4=1
Answer: – 3.5; 1.
So let’s imagine the solution of complete quadratic equations using the diagram in Figure 1.
Using these formulas you can solve any complete quadratic equation. You just need to be careful to the equation was written as a polynomial of the standard form
A x 2 + bx + c, otherwise you may make a mistake. For example, in writing the equation x + 3 + 2x 2 = 0, you can mistakenly decide that
a = 1, b = 3 and c = 2. Then
D = 3 2 – 4 1 2 = 1 and then the equation has two roots. And this is not true. (See solution to example 2 above).
Therefore, if the equation is not written as a polynomial of the standard form, first the complete quadratic equation must be written as a polynomial of the standard form (the monomial with the largest exponent should come first, that is A x 2 , then with less – bx and then a free member With.
When solving the reduced quadratic equation and a quadratic equation with an even coefficient in the second term, you can use other formulas. Let's get acquainted with these formulas. If in a complete quadratic equation the second term has an even coefficient (b = 2k), then you can solve the equation using the formulas shown in the diagram in Figure 2.
A complete quadratic equation is called reduced if the coefficient at x 2 is equal to one and the equation takes the form x 2 + px + q = 0. Such an equation can be given for solution, or it can be obtained by dividing all coefficients of the equation by the coefficient A, standing at x 2 .
Figure 3 shows a diagram for solving the reduced square 
equations. Let's look at an example of the application of the formulas discussed in this article.
Example. Solve the equation
3x 2 + 6x – 6 = 0.
Let's solve this equation using the formulas shown in the diagram in Figure 1.
D = 6 2 – 4 3 (– 6) = 36 + 72 = 108
√D = √108 = √(36 3) = 6√3
x 1 = (-6 - 6√3)/(2 3) = (6 (-1- √(3)))/6 = –1 – √3
x 2 = (-6 + 6√3)/(2 3) = (6 (-1+ √(3)))/6 = –1 + √3
Answer: –1 – √3; –1 + √3
You can notice that the coefficient of x in this equation even number, that is, b = 6 or b = 2k, whence k = 3. Then let’s try to solve the equation using the formulas given in the diagram of the figure D 1 = 3 2 – 3 (– 6) = 9 + 18 = 27
√(D 1) = √27 = √(9 3) = 3√3
x 1 = (-3 - 3√3)/3 = (3 (-1 - √(3)))/3 = – 1 – √3
x 2 = (-3 + 3√3)/3 = (3 (-1 + √(3)))/3 = – 1 + √3
Answer: –1 – √3; –1 + √3. Noticing that all the coefficients in this quadratic equation are divisible by 3 and performing the division, we get the reduced quadratic equation x 2 + 2x – 2 = 0 Solve this equation using the formulas for the reduced quadratic 
equations figure 3.
D 2 = 2 2 – 4 (– 2) = 4 + 8 = 12
√(D 2) = √12 = √(4 3) = 2√3
x 1 = (-2 - 2√3)/2 = (2 (-1 - √(3)))/2 = – 1 – √3
x 2 = (-2 + 2√3)/2 = (2 (-1+ √(3)))/2 = – 1 + √3
Answer: –1 – √3; –1 + √3.
As you can see, when solving this equation using different formulas, we received the same answer. Therefore, having thoroughly mastered the formulas shown in the diagram in Figure 1, you will always be able to solve any complete quadratic equation.
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Formulas for the roots of a quadratic equation. The cases of real, multiple and complex roots are considered. Factorization quadratic trinomial. Geometric interpretation. Examples of determining roots and factoring.
ContentSee also: Solving quadratic equations online
Basic formulas
Consider the quadratic equation:
(1)
.
Roots of a quadratic equation(1) are determined by the formulas:
;
.
These formulas can be combined like this:
.
When the roots of a quadratic equation are known, then a polynomial of the second degree can be represented as a product of factors (factored):
.
We further assume that - real numbers.
Let's 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 quadratic trinomial has the form:
.
If the discriminant is equal to 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 you build graph of a function
,
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 intersects the x-axis (axis) at two points ().
When , the graph touches the x-axis at one point ().
When , the graph does not intersect 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 carry out transformations and apply formulas (f.1) and (f.3):
,
Where
;
.
So, we got the formula for a polynomial of the second degree in the form:
.
This shows 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:
.
We find the discriminant:
.
Since the discriminant is positive, the equation has two real roots:
;
;
.
From here we obtain the factorization of the quadratic trinomial:
.
Graph of the function y = 2 x 2 + 7 x + 3 intersects the x-axis at two points.
Let's plot the function
.
The graph of this function is a parabola. It crosses the abscissa 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)
.
Let's write the quadratic equation in general form:
.
Comparing with the original equation (2.1), we find the values of the coefficients:
.
We find 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). Because this root is factored twice:
,
then such a root is usually called a multiple. That is, they believe that there are two equal roots:
.
;
.
Example 3
Find the roots of a quadratic equation:
(3.1)
.
Let's write the quadratic equation in general form:
(1)
.
Let's rewrite the original equation (3.1):
.
Comparing with (1), we find the values of the coefficients:
.
We find 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 intersect the x-axis (axis). Therefore there are no real roots.
There are no real roots. Complex roots:
;
;
.
", that is, equations of the first degree. In this lesson we will look at what is called 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 power in which the unknown is “2”, then you have a quadratic equation.
Examples of quadratic equations
- 5x 2 − 14x + 17 = 0
- −x 2 + x +
= 01 3 - x 2 + 0.25x = 0
- x 2 − 8 = 0
Important! The general form of a quadratic equation looks like this:
A x 2 + b x + c = 0
“a”, “b” and “c” are given numbers.- “a” is the first or highest coefficient;
- “b” is the second coefficient;
- “c” is a free term.
To find “a”, “b” and “c” you need to compare your equation with the general form of the quadratic equation “ax 2 + bx + c = 0”.
Let's practice determining the coefficients "a", "b" and "c" in quadratic equations.
| 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, a special method is used to solve quadratic equations. formula for finding roots.
Remember!
To solve a quadratic equation you need:
- reduce the quadratic equation to general appearance"ax 2 + bx + c = 0". That is, only “0” should remain on the right side;
- use formula for roots:
Let's look at an example of how to use the formula to find the roots of a quadratic equation. Let's solve a 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 just need to apply formula for finding the roots of a quadratic equation.
Let us determine the coefficients “a”, “b” and “c” for this equation.
x 1;2 =
x 1;2 =
x 1;2 =
x 1;2 =
It can be used to solve any quadratic equation.
In the formula “x 1;2 = ” the radical expression is often replaced
“b 2 − 4ac” for the letter “D” and is called discriminant. The concept of a discriminant is discussed in more detail in the lesson “What is a discriminant”.
Let's look at another example of a quadratic equation.
x 2 + 9 + x = 7x
In this form, it is quite difficult to determine the coefficients “a”, “b” and “c”. Let's first reduce the equation to the general form “ax 2 + bx + c = 0”.
X 2 + 9 + x = 7x
x 2 + 9 + x − 7x = 0
x 2 + 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 quadratic equations have no roots. This situation occurs when the formula contains a negative number under the root.
First, what is a quadratic equation? A quadratic equation is an equation of the form ax^2+bx+c=0, where x is a variable, a, b and c are some numbers, and a is not equal to zero.
Step 2
To solve a quadratic equation, we need to know the formula of its roots, that is, for starters, the discriminant formula of the quadratic equation. It looks like this: D=b^2-4ac. You can derive it yourself, but usually this is not required, just remember the formula (!) You will really need it in the future. There is also a formula for the quarter discriminant, more about it a little later.
Step 3
Let's take as an example the equation 3x^2-24x+21=0. I'll solve it in two ways.
Step 4
Method 1. Discriminant.
3x^2-24x+21=0
a=3, b=-24, c=21
D=b^2-4ac
D=576-4*63=576-252=324=18^2
D>
x1.2= (-b 18)/6=42/6=7
x2=(-(-24)-18)/6=6/6=1
Step 5
It's time to remember the quarter discriminant formula, which can greatly facilitate the solution of our equation =) so, here's what it looks like: D1=k^2-ac (k=1/2b)
Method 2. Quarter Discriminant.
3x^2-24x+21=0
a=3, b=-24, c=21
k=-12
D1=k^2 – ac
D1=144-63=81=9^2
D1>0, which means the equation has 2 roots
x1,2= k +/ square root from D1)/a
x1= (-(-12) +9)/3=21/3=7
x2= (-(-12) -9)/3=3/3=1
Did you rate how much easier the solution is? ;)
Thank you for your attention, I wish you success in your studies =)
- In our case, in the equations D and D1 were >0 and we got 2 roots each. If there were D=0 and D1=0, then we would get one root each, and if there were D<0 и D1<0 соответственно, то у уравнений корней бы не было вовсе.
- Using the root of the discriminant (D1), you can solve only those equations in which the term b is even(!)
