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Quadratic Equations: Solving Quadratic Equations by Factoring

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra</link> by Denny Burzynski and Wade Ellis, Jr. Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method. Objectives of this module: be able to solve quadratic equations by factoring.

Overview

  • Factoring Method
  • Solving Mentally After Factoring

Factoring Method

To solve quadratic equations by factoring, we must make use of the zero-factor property.

  1. Set the equation equal to zero, that is, get all the nonzero terms on one side of the equal sign and 0 on the other.

     a x 2 +bx+c=0 a x 2 +bx+c=0
  2. Factor the quadratic expression.

     ( )( )=0 ( )( )=0
  3. By the zero-factor property, at least one of the factors must be zero, so, set each of the factors equal to 0 and solve for the variable.

Sample Set A

Solve the following quadratic equations. (We will show the check for problem 1.)

Example 1

x 2 −7x+12 = 0. The equation is already  set equal to 0. Factor. ( x−3 )( x−4 ) = 0 Set each factor equal to 0. x−3 = 0 or x−4 = 0 x = 3 or x = 4 x 2 −7x+12 = 0. The equation is already  set equal to 0. Factor. ( x−3 )( x−4 ) = 0 Set each factor equal to 0. x−3 = 0 or x−4 = 0 x = 3 or x = 4
Check: If x=3,  x 2 −7x + 12 = 0 3 2 −7 · 3 + 12 = 0 Is this correct? 9−21 + 12 = 0 Is this correct? 0 = 0 Yes, this is correct. Check: If x=3,  x 2 −7x + 12 = 0 3 2 −7 · 3 + 12 = 0 Is this correct? 9−21 + 12 = 0 Is this correct? 0 = 0 Yes, this is correct.

Check: If x=4,  x 2 −7x + 12 = 0 4 2 −7 · 4 + 12 = 0 Is this correct? 16−28 + 12 = 0 Is this correct? 0 = 0 Yes, this is correct. Check: If x=4,  x 2 −7x + 12 = 0 4 2 −7 · 4 + 12 = 0 Is this correct? 16−28 + 12 = 0 Is this correct? 0 = 0 Yes, this is correct.
Thus, the solutions to this equation are x=3, 4. x=3, 4.

Example 2

x 2 = 25. Set the equation equal to 0. x 2 −25 = 0 Factor. ( x+5 )( x−5 ) = 0 Set each factor equal to 0. x+5=0 or x−5=0 x=−5 or x=5 x 2 = 25. Set the equation equal to 0. x 2 −25 = 0 Factor. ( x+5 )( x−5 ) = 0 Set each factor equal to 0. x+5=0 or x−5=0 x=−5 or x=5
Thus, the solutions to this equation are x=5,−5. x=5,−5.

Example 3

x 2 = 2x. Set the equation equal to 0. x 2 −2x = 0 Factor. x( x−2 ) Set each factor equal to 0. x=0 or x−2=0 x=2 x 2 = 2x. Set the equation equal to 0. x 2 −2x = 0 Factor. x( x−2 ) Set each factor equal to 0. x=0 or x−2=0 x=2
Thus, the solutions to this equation are x=0, 2. x=0, 2.

Example 4

2 x 2 +7x−15 = 0. Factor. ( 2x−3 )( x+5 ) = 0 Set each factor equal to 0. 2x−3=0 or x+5=0 2x=3 or x=−5 x= 3 2 2 x 2 +7x−15 = 0. Factor. ( 2x−3 )( x+5 ) = 0 Set each factor equal to 0. 2x−3=0 or x+5=0 2x=3 or x=−5 x= 3 2
Thus, the solutions to this equation are x= 3 2 ,−5. x= 3 2 ,−5.

Example 5

63 x 2 =13x+6 63 x 2 =13x+6
63 x 2 −13x−6 = 0 ( 9x+2 )( 7x−3 ) = 0 9x+2=0 or 7x−3=0 9x=−2 or 7x=3 x= −2 9 or x= 3 7 63 x 2 −13x−6 = 0 ( 9x+2 )( 7x−3 ) = 0 9x+2=0 or 7x−3=0 9x=−2 or 7x=3 x= −2 9 or x= 3 7
Thus, the solutions to this equation are x= −2 9 , 3 7 . x= −2 9 , 3 7 .

Practice Set A

Solve the following equations, if possible.

Exercise 1

( x−7 )( x+4 )=0 ( x−7 )( x+4 )=0

Solution

x=7, −4 x=7, −4

Exercise 2

( 2x+5 )( 5x−7 )=0 ( 2x+5 )( 5x−7 )=0

Solution

x= −5 2 , 7 5 x= −5 2 , 7 5

Exercise 3

x 2 +2x−24=0 x 2 +2x−24=0

Solution

x=4, −6 x=4, −6

Exercise 4

6 x 2 +13x−5=0 6 x 2 +13x−5=0

Solution

x= 1 3 , −5 2 x= 1 3 , −5 2

Exercise 5

5 y 2 +2y=3 5 y 2 +2y=3

Solution

y= 3 5 ,−1 y= 3 5 ,−1

Exercise 6

m( 2m−11 )=0 m( 2m−11 )=0

Solution

m=0, 11 2 m=0, 11 2

Exercise 7

6 p 2 =−( 5p+1 ) 6 p 2 =−( 5p+1 )

Solution

p= −1 3 , −1 2 p= −1 3 , −1 2

Exercise 8

r 2 −49=0 r 2 −49=0

Solution

r=7,−7 r=7,−7

Solving Mentally After Factoring

Let’s consider problems 4 and 5 of Sample Set A in more detail. Let’s look particularly at the factorizations ( 2x−3 )( x+5 )=0 ( 2x−3 )( x+5 )=0 and ( 9x+2 )( 7x−3 )=0. ( 9x+2 )( 7x−3 )=0. The next step is to set each factor equal to zero and solve. We can solve mentally if we understand how to solve linear equations: we transpose the constant from the variable term and then divide by the coefficient of the variable.

Sample Set B

Example 6

Solve the following equation mentally.

( 2x−3 )( x+5 )=0 ( 2x−3 )( x+5 )=0
2x−3 = 0 Mentally add 3 to both sides. The constant changes sign. 2x = 3 Divide by 2, the coefficient of x. The 2 divides the constant 3 into  3 2 .  The coefficient becomes the denominator. x = 3 2 x+5 = 0 Mentally subtract 5 from both sides. The constant changes sign. x = −5 Divide by the coefficient of  x, 1.The coefficient becomes the denominator. x= −5 1 = −5 x = −5 2x−3 = 0 Mentally add 3 to both sides. The constant changes sign. 2x = 3 Divide by 2, the coefficient of x. The 2 divides the constant 3 into  3 2 .  The coefficient becomes the denominator. x = 3 2 x+5 = 0 Mentally subtract 5 from both sides. The constant changes sign. x = −5 Divide by the coefficient of  x, 1.The coefficient becomes the denominator. x= −5 1 = −5 x = −5
Now, we can immediately write the solution to the equation after factoring by looking at each factor, changing the sign of the constant, then dividing by the coefficient.

Practice Set B

Exercise 9

Solve ( 9x+2 )( 7x−3 )=0 ( 9x+2 )( 7x−3 )=0 using this mental method.

Solution

x=− 2 9 , 3 7 x=− 2 9 , 3 7

Exercises

For the following problems, solve the equations, if possible.

Exercise 10

( x+1 )( x+3 )=0 ( x+1 )( x+3 )=0

Solution

x=−1,  −3 x=−1,  −3

Exercise 11

( x+4 )( x+9 )=0 ( x+4 )( x+9 )=0

Exercise 12

( x−5 )( x−1 )=0 ( x−5 )( x−1 )=0

Solution

x=1,  5 x=1,  5

Exercise 13

( x−6 )( x−3 )=0 ( x−6 )( x−3 )=0

Exercise 14

( x−4 )( x+2 )=0 ( x−4 )( x+2 )=0

Solution

x=−2,  4 x=−2,  4

Exercise 15

( x+6 )( x−1 )=0 ( x+6 )( x−1 )=0

Exercise 16

( 2x+1 )( x−7 )=0 ( 2x+1 )( x−7 )=0

Solution

x=− 1 2 ,  7 x=− 1 2 ,  7

Exercise 17

( 3x+2 )( x−1 )=0 ( 3x+2 )( x−1 )=0

Exercise 18

( 4x+3 )( 3x−2 )=0 ( 4x+3 )( 3x−2 )=0

Solution

x=− 3 4 ,   2 3 x=− 3 4 ,   2 3

Exercise 19

( 5x−1 )( 4x+7 )=0 ( 5x−1 )( 4x+7 )=0

Exercise 20

( 6x+5 )( 9x−4 )=0 ( 6x+5 )( 9x−4 )=0

Solution

x=− 5 6 ,   4 9 x=− 5 6 ,   4 9

Exercise 21

( 3a+1 )( 3a−1 )=0 ( 3a+1 )( 3a−1 )=0

Exercise 22

x( x+4 )=0 x( x+4 )=0

Solution

x=−4,  0 x=−4,  0

Exercise 23

y( y−5 )=0 y( y−5 )=0

Exercise 24

y( 3y−4 )=0 y( 3y−4 )=0

Solution

y=0,    4 3 y=0,    4 3

Exercise 25

b( 4b+5 )=0 b( 4b+5 )=0

Exercise 26

x( 2x+1 )( 2x+8 )=0 x( 2x+1 )( 2x+8 )=0

Solution

x=−4,  −  1 2 ,  0 x=−4,  −  1 2 ,  0

Exercise 27

y( 5y+2 )( 2y−1 )=0 y( 5y+2 )( 2y−1 )=0

Exercise 28

( x−8 ) 2 =0 ( x−8 ) 2 =0

Solution

x=8 x=8

Exercise 29

( x−2 ) 2 =0 ( x−2 ) 2 =0

Exercise 30

( b+7 ) 2 =0 ( b+7 ) 2 =0

Solution

b=−7 b=−7

Exercise 31

( a+1 ) 2 =0 ( a+1 ) 2 =0

Exercise 32

x ( x−4 ) 2 =0 x ( x−4 ) 2 =0

Solution

x=0,  4 x=0,  4

Exercise 33

y ( y+9 ) 2 =0 y ( y+9 ) 2 =0

Exercise 34

y ( y−7 ) 2 =0 y ( y−7 ) 2 =0

Solution

y=0,  7 y=0,  7

Exercise 35

y ( y+5 ) 2 =0 y ( y+5 ) 2 =0

Exercise 36

x 2 −4=0 x 2 −4=0

Solution

x=−2,  2 x=−2,  2

Exercise 37

x 2 +9=0 x 2 +9=0

Exercise 38

x 2 +36=0 x 2 +36=0

Solution

no solution

Exercise 39

x 2 −25=0 x 2 −25=0

Exercise 40

a 2 −100=0 a 2 −100=0

Solution

a=−10,  10 a=−10,  10

Exercise 41

a 2 −81=0 a 2 −81=0

Exercise 42

b 2 −49=0 b 2 −49=0

Solution

b=7,  −7 b=7,  −7

Exercise 43

y 2 −1=0 y 2 −1=0

Exercise 44

3 a 2 −75=0 3 a 2 −75=0

Solution

a=5,  −5 a=5,  −5

Exercise 45

5 b 2 −20=0 5 b 2 −20=0

Exercise 46

y 3 −y=0 y 3 −y=0

Solution

y=0,  1,  −1 y=0,  1,  −1

Exercise 47

a 2 =9 a 2 =9

Exercise 48

b 2 =4 b 2 =4

Solution

b=2,  −2 b=2,  −2

Exercise 49

b 2 =1 b 2 =1

Exercise 50

a 2 =36 a 2 =36

Solution

a=6,  −6 a=6,  −6

Exercise 51

3 a 2 =12 3 a 2 =12

Exercise 52

−2 x 2 =−4 −2 x 2 =−4

Solution

x= 2 ,  − 2 x= 2 ,  − 2

Exercise 53

−2 a 2 =−50 −2 a 2 =−50

Exercise 54

−7 b 2 =−63 −7 b 2 =−63

Solution

b=3,  −3 b=3,  −3

Exercise 55

−2 x 2 =−32 −2 x 2 =−32

Exercise 56

3 b 2 =48 3 b 2 =48

Solution

b=4,  −4 b=4,  −4

Exercise 57

a 2 −8a+16=0 a 2 −8a+16=0

Exercise 58

y 2 +10y+25=0 y 2 +10y+25=0

Solution

y=−5 y=−5

Exercise 59

y 2 +9y+16=0 y 2 +9y+16=0

Exercise 60

x 2 −2x−1=0 x 2 −2x−1=0

Solution

no solution

Exercise 61

a 2 +6a+9=0 a 2 +6a+9=0

Exercise 62

a 2 +4a+4=0 a 2 +4a+4=0

Solution

a=−2 a=−2

Exercise 63

x 2 +12x=−36 x 2 +12x=−36

Exercise 64

b 2 −14b=−49 b 2 −14b=−49

Solution

b=7 b=7

Exercise 65

3 a 2 +18a+27=0 3 a 2 +18a+27=0

Exercise 66

2 m 3 +4 m 2 +2m=0 2 m 3 +4 m 2 +2m=0

Solution

m=0,  −1 m=0,  −1

Exercise 67

3m n 2 −36mn+36m=0 3m n 2 −36mn+36m=0

Exercise 68

a 2 +2a−3=0 a 2 +2a−3=0

Solution

a=−3,  1 a=−3,  1

Exercise 69

a 2 +3a−10=0 a 2 +3a−10=0

Exercise 70

x 2 +9x+14=0 x 2 +9x+14=0

Solution

x=−7,  −2 x=−7,  −2

Exercise 71

x 2 −7x+12=3 x 2 −7x+12=3

Exercise 72

b 2 +12b+27=0 b 2 +12b+27=0

Solution

b=−9,  −3 b=−9,  −3

Exercise 73

b 2 −3b+2=0 b 2 −3b+2=0

Exercise 74

x 2 −13x=−42 x 2 −13x=−42

Solution

x=6,  7 x=6,  7

Exercise 75

a 3 =−8 a 2 −15a a 3 =−8 a 2 −15a

Exercise 76

6 a 2 +13a+5=0 6 a 2 +13a+5=0

Solution

a=− 5 3 ,  − 1 2 a=− 5 3 ,  − 1 2

Exercise 77

6 x 2 −4x−2=0 6 x 2 −4x−2=0

Exercise 78

12 a 2 +15a+3=0 12 a 2 +15a+3=0

Solution

a=− 1 4 ,  −1 a=− 1 4 ,  −1

Exercise 79

18 b 2 +24b+6=0 18 b 2 +24b+6=0

Exercise 80

12 a 2 +24a+12=0 12 a 2 +24a+12=0

Solution

a=−1 a=−1

Exercise 81

4 x 2 −4x=−1 4 x 2 −4x=−1

Exercise 82

2 x 2 =x+15 2 x 2 =x+15

Solution

x=− 5 2 ,  3 x=− 5 2 ,  3

Exercise 83

4 a 2 =4a+3 4 a 2 =4a+3

Exercise 84

4 y 2 =−4y−2 4 y 2 =−4y−2

Solution

no solution

Exercise 85

9 y 2 =9y+18 9 y 2 =9y+18

Exercises For Review

Exercise 86

((Reference)) Simplify ( x 4 y 3 ) 2 ( x y 2 ) 4 . ( x 4 y 3 ) 2 ( x y 2 ) 4 .

Solution

x 12 y 14 x 12 y 14

Exercise 87

((Reference)) Write ( x −2 y 3 w 4 ) −2 ( x −2 y 3 w 4 ) −2 so that only positive exponents appear.

Exercise 88

((Reference)) Find the sum: x x 2 −x−2 + 1 x 2 −3x+2 . x x 2 −x−2 + 1 x 2 −3x+2 .

Solution

x 2 +1 ( x+1 )( x−1 )( x−2 ) x 2 +1 ( x+1 )( x−1 )( x−2 )

Exercise 89

((Reference)) Simplify 1 a + 1 b 1 a − 1 b . 1 a + 1 b 1 a − 1 b .

Exercise 90

((Reference)) Solve ( x+4 )( 3x+1 )=0. ( x+4 )( 3x+1 )=0.

Solution

x=−4,   −1 3 x=−4,   −1 3

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