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Watermelon drop A watermelon is dropped from the tenth story of a building. Solve the equation −16 t 2 + 144 = 0 for t to find the number of seconds it takes the watermelon to reach the ground.

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Writing exercises

Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?

Answers may vary.

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Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has the following statements all to be preceded by “I can…”. The first row is “solve quadratic equations by using the zero product property”. The second row is “solve quadratic equations by factoring”. The third row is “solve applications modeled by quadratic equations”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

7.1 Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

24 x 42

6 ( 4 x 7 )

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15 m 4 + 6 m 2 n

3 m 2 ( 5 m 2 + 2 n )

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Factor by Grouping

In the following exercises, factor by grouping.

a x a y + b x b y

( a + b ) ( x y )

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x 2 y x y 2 + 2 x 2 y

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x 2 + 7 x 3 x 21

( x 3 ) ( x + 7 )

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4 x 2 16 x + 3 x 12

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m 3 + m 2 + m + 1

( m 2 + 1 ) ( m + 1 )

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7.2 Factor Trinomials of the form x 2 + b x + c

Factor Trinomials of the Form x 2 + b x + c

In the following exercises, factor each trinomial of the form x 2 + b x + c .

u 2 + 17 u + 72

( u + 8 ) ( u + 9 )

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k 2 16 k + 60

( k 6 ) ( k 10 )

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y 2 + 6 y 7

( y + 7 ) ( y 1 )

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s 2 2 s 8

( s 4 ) ( s + 2 )

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Factor Trinomials of the Form x 2 + b x y + c y 2

In the following examples, factor each trinomial of the form x 2 + b x y + c y 2 .

x 2 + 12 x y + 35 y 2

( x + 5 y ) ( x + 7 y )

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a 2 + 4 a b 21 b 2

( a + 7 b ) ( a 3 b )

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p 2 5 p q 36 q 2

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7.3 Factoring Trinomials of the form a x 2 + b x + c

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

y 2 17 y + 42

Undo FOIL

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8 a 3 + 72 a

Factor the GCF

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4 m m n 3 n + 12

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Factor Trinomials of the Form a x 2 + b x + c with a GCF

In the following exercises, factor completely.

6 x 2 + 42 x + 60

6 ( x + 2 ) ( x + 5 )

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3 n 4 12 n 3 96 n 2

3 n 2 ( n 8 ) ( n + 4 )

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Factor Trinomials Using the “ac” Method

In the following exercises, factor.

2 x 2 + 9 x + 4

( x + 4 ) ( 2 x + 1 )

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18 a 2 9 a + 1

( 3 a 1 ) ( 6 a 1 )

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15 p 2 + 2 p 8

( 5 p + 4 ) ( 3 p 2 )

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40 s 2 s 6

( 5 s 2 ) ( 8 s + 3 )

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Factor Trinomials with a GCF Using the “ac” Method

In the following exercises, factor.

3 x 2 + 3 x 36

3 ( x + 4 ) ( x 3 )

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60 y 2 85 y 25

5 ( 4 y + 1 ) ( 3 y 5 )

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7.4 Factoring Special Products

Factor Perfect Square Trinomials

In the following exercises, factor.

25 x 2 + 30 x + 9

( 5 x + 3 ) 2

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36 a 2 84 a b + 49 b 2

( 6 a 7 b ) 2

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64 r 2 176 r s + 121 s 2

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40 x 2 + 360 x + 810

10 ( 2 x + 9 ) 2

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2 y 3 16 y 2 + 32 y

2 y ( y 4 ) 2

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5 k 3 70 k 2 + 245 k

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Factor Differences of Squares

In the following exercises, factor.

81 r 2 25

( 9 r 5 ) ( 9 r + 5 )

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169 m 2 n 2

( 13 m + n ) ( 13 m n )

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25 p 2 1

( 5 p 1 ) ( 5 p + 1 )

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9 121 y 2

( 3 + 11 y ) ( 3 11 y )

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20 x 2 125

5 ( 2 x 5 ) ( 2 x + 5 )

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49 u 3 9 u

u ( 7 u + 3 ) ( 7 u 3 )

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Factor Sums and Differences of Cubes

In the following exercises, factor.

a 3 125

( a 5 ) ( a 2 + 5 a + 25 )

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2 m 3 + 54

2 ( m + 3 ) ( m 2 3 m + 9 )

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7.5 General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

24 x 3 + 44 x 2

4 x 2 ( 6 x + 11 )

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16 n 2 56 m n + 49 m 2

( 4 n 7 m ) 2

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5 r 2 + 22 r 48

( r + 6 ) ( 5 r 8 )

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n 4 81

( n 2 + 9 ) ( n + 3 ) ( n 3 )

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5 x 2 + 5 x 60

5 ( x 3 ) ( x + 4 )

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m 3 + 125

( m + 5 ) ( m 2 5 m + 25 )

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2 b 2 2 b c + 5 c b 5 c 2

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7.6 Quadratic Equations

Use the Zero Product Property

In the following exercises, solve.

( a 3 ) ( a + 7 ) = 0

a = 3 a = −7

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( b 3 ) ( b + 10 ) = 0

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3 m ( 2 m 5 ) ( m + 6 ) = 0

m = 0 m = −3 m = 5 2

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7 n ( 3 n + 8 ) ( n 5 ) = 0

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Solve Quadratic Equations by Factoring

In the following exercises, solve.

x 2 + 9 x + 20 = 0

x = −4 , x = −5

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2 p 2 11 p = 40

p = 5 2 , p = 8

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144 m 2 25 = 0

m = 5 12 , m = 5 12

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Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive numbers is 462 . Find the numbers.

−21 , −22 21 , 22

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The area of a rectangular shaped patio 400 square feet. The length of the patio is 9 feet more than its width. Find the length and width.

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Practice test

In the following exercises, find the Greatest Common Factor in each expression.

14 y 42

7 ( y 6 )

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80 a 2 + 120 a 3

40 a 2 ( 2 + 3 a )

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5 m ( m 1 ) + 3 ( m 1 )

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In the following exercises, factor completely.

x 2 + 13 x + 36

( x + 7 ) ( x + 6 )

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3 a 3 6 a 2 72 a

3 a ( a 2 2 a 14 )

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5 n 2 + 30 n + 45

5 ( n + 1 ) ( n + 5 )

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x y 8 y + 7 x 56

( x 8 ) ( y + 7 )

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9 s 2 12 s + 4

( 3 s 2 ) 2

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100 a 2

( 10 a ) ( 10 + a )

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3 x 2 75 y 2

3 ( x + 5 y ) ( x 5 y )

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a b 3 b 2 a + 6

( a 3 ) ( b 2 )

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8 m 2 + 22 m + 5

( 4 m + 1 ) ( 2 m + 5 )

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In the following exercises, solve.

y 2 = y + 132

y = −11 , y = 12

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9 b 2 9 = 0

b = 1 , b = −1

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4 n 2 + 19 + 21 = 0

n = 7 4 , n = −3

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( x 3 ) ( x + 2 ) = 6

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The product of two consecutive integers is 156 . Find the integers.

12 and 13 ; 13 and −12

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The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.

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Practice Key Terms 2

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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