Transcript Lesson 5.3 Factoring Quadratic Expressions

5 minutes
Warm-Up
List all the factors of each number.
1) 10
2) 48
3) 7
Find the greatest common factor (GCF) of each set of
numbers.
4) 6,14
5) 12,18,30
6) 4,8,15,20
5.3.1 Factoring Quadratic Expressions
Objectives:
•Factor a quadratic expression
Example 1
Factor each quadratic expression.
a) 27
27x
x2 – 18x
18x
9 x (3x – 2)
b) 5x(2x
(2x + 1) – 2(2x
(2x + 1)
(2x + 1)( 5x - 2 )
factor out the
GCF for all terms
factor out the
GCF for all terms
Factoring x2 + bx + c
To factor an expression of the form ax2 + bx + c,
where a = 1, look for integers r and s such that r s = c
and r + s = b. Then factor the expression.
x2 + bx + c = (x + r)(x + s)
Example 2
Factor x2 + 12x + 27.
(x + 3)(x + 9)
Example 3
Factor x2 - 15x - 54.
(x + 3)(x - 18)
Example 4
Factor 5x2 + 14x + 8.
( 5x + 4) (x + 2)
Practice
Factor.
1) 5x2 + 15x
2) (2x – 1)4 + (2x – 1)x
3) x2 + 9x + 20
4) x2 – 7x - 30
5) 3x2 + 11x - 20
Homework
p.296 #31,35,37,39,41,43,45,49,53,57
5 minutes
Warm-Up
Factor.
1) 3x2 - 15x
2) (3x + 7)x + (3x + 7)8
3) x2 + 14x + 49
4) x2 – 13x - 30
5) 7x2 – 11x - 6
5.3.2 Factoring Quadratic Expressions
Objectives:
•Factor a quadratic expression
•Use factoring to solve a quadratic equation and find
the zeros of a quadratic function
Special Products
Factoring the Difference of Two Squares
a2 – b2 = (a + b)(a – b)
Factoring Perfect-Square Trinomials
a2 + 2ab + b2 = (a + b)(a + b)
a2 - 2ab + b2 = (a - b)(a - b)
Example 1
Factor x2 - 16.
(x + 4) (x - 4)
Example 2
Factor x4 - 81.
(x2+ 9) (x2 - 9)
(x2 + 9) (x + 3 ) ( x - 3)
Example 3
Factor 2x
2 2 – 24x
24 + 72
72.
2( x2 – 12x + 36 )
2(x - 6)(x - 6)
Zero-Product Property
If pq = 0, then p = 0 or q = 0.
Example 4
Solve 5x2 + 7x = 0.
x(5x + 7) = 0
x=0
or
5x + 7 = 0
5x = -7
7
x
5
CHECK: 5x2 + 7x = 0
5(0)2 + 7(0) = 0
0+0=0
CHECK:
5x2 + 7x = 0
2
 7
 7
5    7    0
 5
 5
49
 49 
5
0

5
 25 
Example 5
Find the zeroes of the function f(x) = x2 – 5x + 6
x2 – 5x + 6 = 0
(x – 3)(x – 2) = 0
x -3 = 0
x=3
CHECK:
or
x2 – 4x = x - 6
32 – 4(3) = 3 - 6
9 – 12 = -3
-3 = -3
x-2=0
x=2
CHECK:
x2 – 4x = x - 6
22 – 4(2) = 2 - 6
4 – 8 = -4
-4 = -4
The zeroes are located at x = 2 and x = 3.
Practice
Find the zeroes of each function.
1) h(x) = 3x2 + 12x
2) j(x) = x2 + 4x - 21
Homework
p.296 #59,61,65,67,69,71,75,79,83