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2. You want (x m) (x n) where mn 6
Example 1
Factor when c is Positive
x 2 bx + c
Factor the expression. a. b.
y 2 6y 8 + –
x 2 + 5x + 6
You want (x m) (x n) where mn 6
and m n Because mn is positive, m and n must have the same sign. Since mn 6, find factors of 6 that have a sum of 5.
SOLUTION =
x 2 5x + 6
Factors of 6: m, n 7 5
1, 6 –
1, 6
2, 3
2, 3
Sum of factors: m + n 2

3. (x 2) (x 3). Check your answer by multiplying.
Example 1
Factor when c is Positive
x 2 bx + c
ANSWER + =
x 2 5x 6
(x ) (x ). Check your answer by
multiplying.
CHECK ( ) 2 + x 3 =
x 2 3x 2x 6 5x 3

4. (y 2) (y 4). Check your answer by multiplying. – y 2 6y 8 +
Example 1
Factor when c is Positive
x 2 bx + c
y 2 6y 8 + –
b. You want (y m)(y n) where mn and m n Because mn is positive, m and n must have the same sign. =
Factors of 8: m, n 9 6
1, 8 –
1, 8
2, 4
2, 4
Sum of factors: m + n
ANSWER =
(y ) (y ). Check your answer by
multiplying. –
y 2 6y 8 + 4

5. You want (x m) (x n) where mn 9
Example 2
Factor when c is Negative
x 2 bx + c
Factor the expression. a. – b.
x 2 + 8x 9
z 2 –
14z – 15
SOLUTION
You want (x m) (x n) where mn
and m n Because mn is negative, m and n must have different signs. =
x 2 8x + 9 –
Factors of 9: m, n
Sum of factors: m + n
1, 9
1, 9 8
3, 3
ANSWER
(x ) (x ). + =
x 2 8x 9 – 5

6. CHECK Check your answer by multiplying.
Example 2
Factor when c is Negative
x 2 bx + c
CHECK Check your answer by multiplying. ( ) 1 x 9 + =
x 2 9x –
Multiply using FOIL. =
x 2 8x + 9 –
Combine like terms.
When you multiply the binomial factors, you obtain the original expression, so the answer is correct. 6

7. You want (z m) (z n) where mn 15 and m n 14. Because mn is negative,
Example 2
Factor when c is Negative
x 2 bx + c
You want (z m) (z n) where
mn and m n Because mn is negative,
m and n must have different signs.
z 2
14z 15 – + =
Factors of 15: m, n
Sum of factors: m + n –
1, 15
1, 15 14
3, 5 2
3, 5
ANSWER
(z ) (z ). Check your answer by
multiplying. –
z 2
14z 15 = + 7

8. Checkpoint Factor the expression. 1. x 2 6x + 5 ANSWER ( ) 1 + x 5 2.bx + c
Factor the expression. 1.
x 2 6x + 5
ANSWER ( ) 1 + x 5 2.
b 2 7b + 12
ANSWER ( ) 3 + b 4 3.
s 2 5s 4 + –
ANSWER ( ) 4 – s 1 4.
y 2
11y 12 + –
ANSWER ( ) 12 + y 1 – 5.
x 2 x + 6 –
ANSWER ( ) 3 + x 2 –

9. Checkpoint Factor the expression. 6. x 2 – 15x – 16 ANSWER ( ) 16 x 1bx + c
Factor the expression. 6.
x 2 –
15x – 16
ANSWER ( ) 16 x 1 + –

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11. Example 3 Solve the equation . = x 2 + 2x 15 SOLUTION = x 2 + 2x 15 =
Solve a Quadratic Equation by Factoring
Solve the equation =
x 2 + 2x 15
SOLUTION
Write original equation. =
x 2 + 2x 15
Write in standard form. =
x 2 + 2x 15 –
Factor. = ( ) 3 x – 5 + = 3 x – or 5 +
Use the zero product property. = 3 x 5 –
Solve for x.
ANSWER
The solutions are 3 and 5. –

12. A group of students from your school
Example 4
Use a Quadratic Equation as a Model
Community Service
A group of students from your school
volunteers to build a neighborhood
playground. The playground will have
a mulch border along two sides. The
mulch border will have the same width
on both sides. The playground is a
rectangle, as shown. The length of the
playground is 20 yards. The width of
the playground is 10 yards. There is
enough mulch to cover 64 square yards
for the border. How wide should the
border be? 12

13. Example 4
Use a Quadratic Equation as a Model
SOLUTION
Use the formula for the area of a rectangle, Area length width. The area of the playground is square yards. The area of the border will be 64 square yards. So, the total area of the border and the playground will be 264 square yards. = •
Formula for area of a rectangle w = • A = ( ) 20 x + 10
264
Substitute x for and x for w.
Multiply using FOIL. =
264
x 2 +
10x
20x
200 =
264
x 2 +
30x
200
Combine like terms. 13

14. Reject 32 as a solution, because a negative width does not make sense.
Example 4
Use a Quadratic Equation as a Model =
x 2 +
30x 64 –
Write in standard form. = ( ) 32 x + 2 –
Factor. = 32 x + or 2 –
Use the zero product property. = 32 x – 2
Solve for x.
Reject as a solution, because a negative width does not make sense. –
ANSWER
The border should be 2 yards wide. 14

15. Checkpoint Solve the equation. 1. = x 2 10x + 9 – ANSWER 9, 1 2. = y 2
Solve a Quadratic Equation by Factoring
Solve the equation. 1. =
x 2
10x + 9 –
ANSWER
9, 1 2. =
y 2 5y + 14
ANSWER
7, 2 – 3. =
x 2 5 – 4x
ANSWER
5, 1 –

16. Your school plans to increase the area of the parking
Parking Lot
Your school plans to increase the area of the parking
lot by 1000 square yards. The original parking lot is a
rectangle, as shown. The length and width of the
parking lot will each increase by x yards. The width
of the original parking lot is 40 yards, and the length
of the original parking lot is 50 yards.
1) Find the area of the original
parking lot.
2) Find the total area of the parking lot with the new space.
3) Write an equation that you can use to find the value of x. 16

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