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Quadratic Equations: Solving Quadratic Equations

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 place a quadratic equation into standard form, be familiar with the zero-factor property of real numbers.

Overview

  • Standard Form of A Quadratic Equation
  • Zero-Factor Property of Real Numbers

Standard Form of A Quadratic Equation

In Chapter ((Reference)) we studied linear equations in one and two variables and methods for solving them. We observed that a linear equation in one variable was any equation that could be written in the form ax+b=0, a≠0, ax+b=0, a≠0, and a linear equation in two variables was any equation that could be written in the form ax+by=c, ax+by=c, where a a and b b are not both 0. We now wish to study quadratic equations in one variable.

Quadratic Equation

A quadratic equation is an equation of the form a x 2 +bx+c=0, a≠0. a x 2 +bx+c=0, a≠0.

The standard form of the quadratic equation is a x 2 +bx+c=0, a≠0. a x 2 +bx+c=0, a≠0.

For a quadratic equation in standard form a x 2 +bx+c=0, a x 2 +bx+c=0,

a a is the coefficient of x 2 . x 2 .
b b is the coefficient of x. x.
c c is the constant term.

Sample Set A

The following are quadratic equations.

Example 1

3 x 2 +2x−1=0. a=3,b=2,c=−1. 3 x 2 +2x−1=0. a=3,b=2,c=−1.

Example 2

5 x 2 +8x=0. a=5,b=8,c=0. 5 x 2 +8x=0. a=5,b=8,c=0.
Notice that this equation could be written 5 x 2 +8x+0=0. 5 x 2 +8x+0=0. Now it is clear that c=0. c=0.

Example 3

x 2 +7=0. a=1,b=0,c=7. x 2 +7=0. a=1,b=0,c=7.
Notice that this equation could be written x 2 +0x+7=0. x 2 +0x+7=0. Now it is clear that b=0. b=0.

The following are not quadratic equations.

Example 4

3x+2=0. a=0. This equation is linear. 3x+2=0. a=0. This equation is linear.

Example 5

8 x 2 + 3 x −5=0. 8 x 2 + 3 x −5=0.
The expression on the left side of the equal sign has a variable in the denominator and, therefore, is not a quadratic.

Practice Set A

Which of the following equations are quadratic equations? Answer “yes” or “no” to each equation.

Exercise 1

6 x 2 −4x+9=0 6 x 2 −4x+9=0

[ Show Solution ]

Exercise 2

5x+8=0 5x+8=0

[ Show Solution ]

Exercise 3

4 x 3 −5 x 2 +x+6=8 4 x 3 −5 x 2 +x+6=8

[ Show Solution ]

Exercise 4

4 x 2 −2x+4=1 4 x 2 −2x+4=1

[ Show Solution ]

Exercise 5

2 x −5 x 2 =6x+4 2 x −5 x 2 =6x+4

[ Show Solution ]

Exercise 6

9 x 2 −2x+6=4 x 2 +8 9 x 2 −2x+6=4 x 2 +8

[ Show Solution ]

Zero-Factor Property

Our goal is to solve quadratic equations. The method for solving quadratic equations is based on the zero-factor property of real numbers. We were introduced to the zero-factor property in Section (Reference). We state it again.

Zero-Factor Property

If two numbers a a and b b are multiplied together and the resulting product is 0, then at least one of the numbers must be 0. Algebraically, if a · b=0, a · b=0, then a=0 a=0 or b=0, b=0, or both a=0 a=0 and b=0. b=0.

Sample Set B

Use the zero-factor property to solve each equation.

Example 6

If 9x=0,then x must be 0. 9x=0,then x must be 0.

Example 7

If −2 x 2 =0, −2 x 2 =0, then x 2 =0, x=0. x 2 =0, x=0.

Example 8

If 5 5( x−1 )=0, then x−1 x−1 must be 0, since 5 is not zero.

 x−1 = 0 x = 1 x−1 = 0 x = 1

Example 9

If x( x+6 )=0, x( x+6 )=0, then

 x = 0 or x+6 = 0 x = −6 x = 0, −6. x = 0 or x+6 = 0 x = −6 x = 0, −6.

Example 10

If ( x+2 )( x+3 )=0, ( x+2 )( x+3 )=0, then

 x+2 = 0 or x+3 = 0 x = −2 x = −3 x = −2, −3. x+2 = 0 or x+3 = 0 x = −2 x = −3 x = −2, −3.

Example 11

If ( x+10 )( 4x−5 )=0, ( x+10 )( 4x−5 )=0, then

 x+10 = 0 or 4x−5 = 0 x = −10 4x = 5 x = 5 4 x = −10, 5 4 . x+10 = 0 or 4x−5 = 0 x = −10 4x = 5 x = 5 4 x = −10, 5 4 .

Practice Set B

Use the zero-factor property to solve each equation.

Exercise 7

6( a−4 )=0 6( a−4 )=0

[ Show Solution ]

Exercise 8

( y+6 )( y−7 )=0 ( y+6 )( y−7 )=0

[ Show Solution ]

Exercise 9

( x+5 )( 3x−4 )=0 ( x+5 )( 3x−4 )=0

[ Show Solution ]

Exercises

For the following problems, write the values of a, b, a, b, and c c in quadratic equations.

Exercise 10

3 x 2 +4x−7=0 3 x 2 +4x−7=0

[ Show Solution ]

Exercise 11

7 x 2 +2x+8=0 7 x 2 +2x+8=0

Exercise 12

2 y 2 −5y+5=0 2 y 2 −5y+5=0

[ Show Solution ]

Exercise 13

7 a 2 +a−8=0 7 a 2 +a−8=0

Exercise 14

−3 a 2 +4a−1=0 −3 a 2 +4a−1=0

[ Show Solution ]

Exercise 15

7 b 2 +3b=0 7 b 2 +3b=0

Exercise 16

2 x 2 +5x=0 2 x 2 +5x=0

[ Show Solution ]

Exercise 17

4 y 2 +9=0 4 y 2 +9=0

Exercise 18

8 a 2 −2a=0 8 a 2 −2a=0

[ Show Solution ]

Exercise 19

6 x 2 =0 6 x 2 =0

Exercise 20

4 y 2 =0 4 y 2 =0

[ Show Solution ]

Exercise 21

5 x 2 −3x+9=4 x 2 5 x 2 −3x+9=4 x 2

Exercise 22

7 x 2 +2x+1=6 x 2 +x−9 7 x 2 +2x+1=6 x 2 +x−9

[ Show Solution ]

Exercise 23

−3 x 2 +4x−1=−4 x 2 −4x+12 −3 x 2 +4x−1=−4 x 2 −4x+12

Exercise 24

5x−7=−3 x 2 5x−7=−3 x 2

[ Show Solution ]

Exercise 25

3x−7=−2 x 2 +5x 3x−7=−2 x 2 +5x

Exercise 26

0= x 2 +6x−1 0= x 2 +6x−1

[ Show Solution ]

Exercise 27

9= x 2 9= x 2

Exercise 28

x 2 =9 x 2 =9

[ Show Solution ]

Exercise 29

0=− x 2 0=− x 2

For the following problems, use the zero-factor property to solve the equations.

Exercise 30

4x=0 4x=0

[ Show Solution ]

Exercise 31

16y=0 16y=0

Exercise 32

9a=0 9a=0

[ Show Solution ]

Exercise 33

4m=0 4m=0

Exercise 34

3( k+7 )=0 3( k+7 )=0

[ Show Solution ]

Exercise 35

8( y−6 )=0 8( y−6 )=0

Exercise 36

−5( x+4 )=0 −5( x+4 )=0

[ Show Solution ]

Exercise 37

−6( n+15 )=0 −6( n+15 )=0

Exercise 38

y( y−1 )=0 y( y−1 )=0

[ Show Solution ]

Exercise 39

a( a−6 )=0 a( a−6 )=0

Exercise 40

n( n+4 )=0 n( n+4 )=0

[ Show Solution ]

Exercise 41

x( x+8 )=0 x( x+8 )=0

Exercise 42

9( a−4 )=0 9( a−4 )=0

[ Show Solution ]

Exercise 43

−2( m+11 )=0 −2( m+11 )=0

Exercise 44

x( x+7 )=0 x( x+7 )=0

[ Show Solution ]

Exercise 45

n( n−10 )=0 n( n−10 )=0

Exercise 46

( y−4 )( y−8 )=0 ( y−4 )( y−8 )=0

[ Show Solution ]

Exercise 47

( k−1 )( k−6 )=0 ( k−1 )( k−6 )=0

Exercise 48

( x+5 )( x+4 )=0 ( x+5 )( x+4 )=0

[ Show Solution ]

Exercise 49

( y+6 )( 2y+1 )=0 ( y+6 )( 2y+1 )=0

Exercise 50

( x−3 )( 5x−6 )=0 ( x−3 )( 5x−6 )=0

[ Show Solution ]

Exercise 51

( 5a+1 )( 2a−3 )=0 ( 5a+1 )( 2a−3 )=0

Exercise 52

( 6m+5 )( 11m−6 )=0 ( 6m+5 )( 11m−6 )=0

[ Show Solution ]

Exercise 53

( 2m−1 )( 3m+8 )=0 ( 2m−1 )( 3m+8 )=0

Exercise 54

( 4x+5 )( 2x−7 )=0 ( 4x+5 )( 2x−7 )=0

[ Show Solution ]

Exercise 55

( 3y+1 )( 2y+1 )=0 ( 3y+1 )( 2y+1 )=0

Exercise 56

( 7a+6 )( 7a−6 )=0 ( 7a+6 )( 7a−6 )=0

[ Show Solution ]

Exercise 57

( 8x+11 )( 2x−7 )=0 ( 8x+11 )( 2x−7 )=0

Exercise 58

( 5x−14 )( 3x+10 )=0 ( 5x−14 )( 3x+10 )=0

[ Show Solution ]

Exercise 59

( 3x−1 )( 3x−1 )=0 ( 3x−1 )( 3x−1 )=0

Exercise 60

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

[ Show Solution ]

Exercise 61

( 7a−2 ) 2 =0 ( 7a−2 ) 2 =0

Exercise 62

( 5m−6 ) 2 =0 ( 5m−6 ) 2 =0

[ Show Solution ]

Exercises For Review

Exercise 63

((Reference)) Factor 12ax−3x+8a−2 12ax−3x+8a−2 by grouping.

Exercise 64

((Reference)) Construct the graph of 6x+10y−60=0. 6x+10y−60=0.
An xy coordinate plane with gridlines, labeled negative five and five with increments of one units on both axes.

[ Show Solution ]

Exercise 65

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

Exercise 66

((Reference)) Simplify 7 ( 2 +2 ). 7 ( 2 +2 ).

[ Show Solution ]

Exercise 67

((Reference)) Solve the radical equation 3x+10 =x+4. 3x+10 =x+4.

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