Find the product
1. (4y – 3)(3y + 8)
12y2
+ 23y – 24
3. (4q – 5)2
16q2 – 40q + 25
2. (5m + 6)(5m – 6)
25m2 – 36
4. Solve x2 – x – 30 = 0.
(x – 6 )(x + 5) = 0
x = 6 or x = -5
EXAMPLE 1
Factor 5x2 – 17x + 6.
(5x – 2 )(x – 3 )
Factor 3x2 + 20x – 7.
Factors of +30
That add up to -17
-15 and -2
Factors of -21
That add up to + 20
(3x – 1 )(x + 7 )
21 and -1
GUIDED PRACTICE
for Examples 1 and 2
Factor the expression. If the expression cannot be factored, say so.
1.
7x2 – 20x – 3
Factors of -21 that add up to -20
5z2 + 16z + 3
2.
Factors of 15 that add up to 16
-21 and 1
15 and 1
(7x + 1 )(x – 3 )
3.
2w2 + w + 3
Factors of 6 that add up to 1
There are none
cannot be factored
(5z + 1 )(z + 3 )
4.
3x2 + 5x – 12
Factors of -36 that add up to 5
9 and -4
(3x – 4 )(x + 3 )
GUIDED PRACTICE
5.
4u2
+ 12u + 5
10 and 2
)(u
4x2 – 9x + 2
6.
Factors of 20 that add up to 12
(4u
for Examples 1 and 2
Factors of 8 that add up to -9
-8 and -1
)
(2u + 1 )(2u + 5 )
(4x – 1 )(x – 2 )
(2x
)(2x
)
Recall: x2 – y2 = (x – y)(x + y)
Example: 4x2 – 25 = (2x – 5)(2x + 5)
Recall: x2 + 2xy + y2 = (x + y)2
Example: 9x2 + 30x + 25 = (3x + 5)2
EXAMPLE 3
Factor with special patterns
Factor the expression.
a. 9x2 – 64 = (3x – 8)(3x + 8)
b. 4y2 + 20y + 25 = (2y + 5)2
c.
36w2 – 12w + 1 = (6w – 1)2
Difference of two
squares
Perfect square
trinomial
Perfect square
trinomial
EXAMPLE 4
Recall: GCF (Greatest Common Factor)
Factor the expression.
a.
5x2 – 45 = 5(x2 – 9)
= 5(x + 3)(x – 3)
b. 6q2 – 14q + 8 = 2(3q2 – 7q + 4)
= 2(3q – 4)(q – 1)
c.
–5z2 + 20z = –5z(z – 4)
d. 12p2 – 21p + 3 = 3(4p2 – 7p + 1)
EXAMPLE 5
Solve quadratic equations
Solve (a) 3x2 + 10x – 8 = 0
a.
3x2 + 10x – 8 = 0
Factors of -24 that add up to 10
12 and -2
(3x – 2 )(x + 4 ) = 0
3x – 2 = 0
or x + 4 = 0
Write original equation.
Factor.
Zero product property
3x = 2
Solve for x.
x= 2
3
or x = –4
EXAMPLE 5
Solve quadratic equations
(b) 5p2 – 16p + 15 = 4p – 5.
b.
5p2 – 16p + 15 = 4p – 5.
5p2 – 20p + 20 = 0
5(p2 – 4p + 4) = 0
p2 – 4p + 4 = 0
(p – 2)2 = 0
p–2=0
p=2
Write original equation.
Write in standard form.
Factor out a 5.
Divide each side by 5.
Factor.
Zero product property
Solve for p.