Download 6 -6 Factoring by Grouping

SAT Problem of the Day
SAT Problem of the Day
5.3 Factoring Quadratic Expressions
Objectives:
•Factor a quadratic expression
•Use factoring to solve a quadratic equation and find
the zeros of a quadratic function
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 a quadratic expression
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)
Factor a quadratic expression
Example 2
Factor x2 + 12x + 27.
(x + 3)(x + 9)
Factor a quadratic expression
Example 3
Factor x2 - 15x - 54.
(x + 3)(x - 18)
Factor a quadratic expression
Example 4
Factor 5x2 + 14x + 8.
( 5x + 4) (x + 2)
Factor a quadratic expression
Practice
Factor.
1) 5x2 + 15x
2) (2x – 1)4 + (2x – 1)x
3) x2 + 9x + 20
4) x2 – 7x - 30
5) 3x2 + 11x - 20
Factor a quadratic expression
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)
Factor a quadratic expression
Example 5
Factor x2 - 16.
(x + 4) (x - 4)
Factor a quadratic expression
Example 6
Factor x4 - 81.
(x2+ 9) (x2 - 9)
(x2 + 9) (x + 3 ) ( x - 3)
Factor a quadratic expression
Example 7
Factor 2x
2 2 – 24x
24 + 72
72.
2( x2 – 12x + 36 )
2(x - 6)(x - 6)
Factor a quadratic expression
Zero-Product Property
If pq = 0, then p = 0 or q = 0.
Find the zeros of a quadratic function
Example 8
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 
Find the zeros of a quadratic function
Example 9
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.
Find the zeros of a quadratic function
Practice
Find the zeroes of each function.
1) h(x) = 3x2 + 12x
2) j(x) = x2 + 4x - 21
Find the zeros of a quadratic function
Collins Type 2
What do you know about the factors of
x2+bx+c when c is positive? When c is
negative? What information does the sign
of b give you in each case?
Factor a quadratic expression
Homework
Do the problems listed on page 5 of today's
packet ("Finding zeros of quadratic
functions using factoring"). Some of these
are exercises from Lesson 5.3 of the
textbook.
Factor a quadratic expression
Find the zeros of a quadratic function