factoring quadratic equations objective: to factor a quadratic expression and to use factoring to...
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Factoring Quadratic Equations
Objective: To factor a quadratic expression and to use factoring to
solve quadratic equations.
Factoring
• We will be factoring many different types of quadratic equations. The first type we will look at is an expression containing two or more terms.
Factoring
• We will be factoring many different types of quadratic equations. The first type we will look at is an expression containing two or more terms.
• The first thing that you will always do is to factor out the greatest common factor.
Example 1
Example 1
Example 1
Try This
• Factor each quadratic expression.
xx 155 2 xxx )12(4)12(
Try This
• Factor each quadratic expression.
)3(5
355
155 2
xx
xxx
xx xxx )12(4)12(
Try This
• Factor each quadratic expression.
)3(5
355
155 2
xx
xxx
xx)4)(12(
)12(4)12(
xx
xxx
Factoring ax2 + bx + c
• We will now look at factoring quadratic expressions where a is 1 and c is positive.
Factoring ax2 + bx + c
• We will now look at factoring quadratic expressions where a is 1 and c is positive.
• We will start each problem by looking at factors of c.
Factoring ax2 + bx + c
• We will now look at factoring quadratic expressions where a is 1 and c is positive.
• We will start each problem by looking at factors of c.• We want these factors to add together to equal b.
Factoring ax2 + bx + c
• We will now look at factoring quadratic expressions where a is 1 and c is positive.
• We will start each problem by looking at factors of c.• We want these factors to add together to equal b.
• We want factors of c that add together to equal b.
Factoring ax2 + bx + c
• We will now look at factoring quadratic expressions where a is 1 and c is positive.
• We will start each problem by looking at factors of c.• We want these factors to add together to equal b.• If b is positive, both terms will be ( x + ) and if b is
negative, both terms will be (x - ).
Factoring
• Factor the following. 1272 xx
Factoring
• Factor the following.
• We are looking for factors of 12 that add together to equal 7.
1272 xx
Factoring
• Factor the following.
• We are looking for factors of 12 that add together to equal 7.
• The factors of 12 are:
1272 xx
43
62
121
Factoring
• Factor the following.
• We are looking for factors of 12 that add together to equal 7.
• The factors of 12 are:
1272 xx
43
62
121
)4)(3( xx
You Try
• Factor the following. 1892 xx
You Try
• Factor the following.
• We are looking for factors of 18 that add together to equal 9.
• The factors of 18 are:
1892 xx
63
92
181
You Try
• Factor the following.
• We are looking for factors of 18 that add together to equal 9.
• The factors of 18 are:
1892 xx
63
92
181
)6)(3( xx
Example
• Factor the following. 32122 xx
Example
• Factor the following.
• We are looking for factors of 32 that add together to equal 12.
• The factors of 32 are:
32122 xx
84
162
321
Example
• Factor the following.
• We are looking for factors of 32 that add together to equal 12.
• The factors of 18 are:
32122 xx
84
162
321
)4)(8( xx
You Try
• Factor the following. 28112 xx
You Try
• Factor the following.
• We are looking for factors of 28 that add together to equal 11.
• The factors of 28 are:
28112 xx
74
142
281
You Try
• Factor the following.
• We are looking for factors of 28 that add together to equal 11.
• The factors of 28 are:
28112 xx
74
142
281
)4)(7( xx
Factoring
• We will now look at expressions where c is negative.
• Now we will say factors of c that are different by b.
• The larger factor has the same sign as b.
Factoring
• Factor the following. 3072 xx
Factoring
• Factor the following.
• We are looking for factors of 30 that are different by 7.
• The factors of 30 are:
3072 xx
65
103
152
301
Factoring
• Factor the following.
• We are looking for factors of 30 that are different by 7.
• The factors of 30 are:
3072 xx
65
103
152
301
)3)(10( xx
You Try
• Factor the following. 2762 xx
You Try
• Factor the following.
• We are looking for factors of 27 that are different by 6.
• The factors of 27 are:
2762 xx
93
271
You Try
• Factor the following.
• We are looking for factors of 27 that are different by 6.
• The factors of 27 are:
2762 xx
93
271
)3)(9( xx
Factoring
• We will now look at factoring when a is not 1. • If c is positive, we are looking for the outer and inner
to add together to equal b.• If c is negative, we are looking for the outer and inner
to be different by b.• We need to start with factors of a. We will always
choose the factors that are closet to each other.
Factoring
• Factor the following. 3116 2 xx
Factoring
• Factor the following.
• We are looking for the outer and inner to add together to be 11.
• We will start with 2 and 3.
3116 2 xx
) 2)( 3( xx
Factoring
• Factor the following.
• We are looking for the outer and inner to add together to be 11.
• We will start with 2 and 3.
• The outer is 3x and the inner is 6x. Do not add to equal 11.
3116 2 xx
)1 2)(3 3( xx
Factoring
• Factor the following.
• We are looking for the outer and inner to add together to be 11.
• We will start with 2 and 3.
• The outer is 9x and the inner is 2x. Does add to equal 11.
3116 2 xx
)3 2)(1 3( xx
Factoring
• Factor the following.
• We are looking for the outer and inner to add together to be 11.
• We will start with 2 and 3.
• The outer is 9x and the inner is 2x. Does add to equal 11.
3116 2 xx
)32)(13( xx
You Try
• Factor the following. 20113 2 xx
You Try
• Factor the following.
• We are looking for the outer and inner to be different by 11.
• We will start with 1 and 3.
20113 2 xx
) 3)( ( xx
You Try
• Factor the following.
• We are looking for the outer and inner to be different by 11.
• We will start with 1 and 3.
• The outer is 5x and the inner is 12x. Is not different by 11.
20113 2 xx
) 5 3)( 4 ( xx
You Try
• Factor the following.
• We are looking for the outer and inner to be different by 11.
• We will start with 1 and 3.
• The outer is 4x and the inner is 15x. Is different by 11.
20113 2 xx
)4 3)(5 ( xx
You Try
• Factor the following.
• We are looking for the outer and inner to be different by 11.
• We will start with 1 and 3.
• The outer is 4x and the inner is 15x. Is different by 11.
20113 2 xx
)4 3)(5 ( xx
)43)(5( xx
The Difference of Two Squares
• The word different means subtraction. When two perfect squares are being subtracted, there is a special way to factor this.
))((22 bababa
The Difference of Two Squares
• Factor the following.
92 x
The Difference of Two Squares
• Factor the following.
92 x
)3)(3( xx
You Try
• Factor the following.
162 x
You Try
• Factor the following.
162 x
)4)(4( xx
You Try
• Factor the following.
254 2 x
You Try
• Factor the following.
254 2 x
)52)(52( xx
Factoring Perfect Squares
• If the first term and last term are both perfect squares, you should see if the trinomial factors as a perfect square.
Factoring Perfect Squares
• If the first term and last term are both perfect squares, you should see if the trinomial factors as a perfect square.
222 )(2 bababa
222 )(2 bababa
Factoring Perfect Squares
• Factor the following.
36122 xx
Factoring Perfect Squares
• Factor the following.
36122 xx
22 )6(3612 xxx
)6)(6(36122 xxxx
Zero Product Property
• If two or more terms are being multiplied together and their product is zero, one of them must equal zero.
.0or 0 then ,0 If qppq
Example 6
Example 6
Example 6
You Try
• Use the zero product property to find the zeros of each function.
xxxf 123)( 2 214)( 2 xxxf
You Try
• Use the zero product property to find the zeros of each function.
xxxf 123)( 2 214)( 2 xxxf
0)4(3
0123 2
xx
xx
You Try
• Use the zero product property to find the zeros of each function.
xxxf 123)( 2 214)( 2 xxxf
0)4(3
0123 2
xx
xx
0
03
x
x
4
04
x
x
You Try
• Use the zero product property to find the zeros of each function.
xxxf 123)( 2 214)( 2 xxxf
0)4(3
0123 2
xx
xx
0
03
x
x
4
04
x
x
0)3)(7(
02142
xx
xx
You Try
• Use the zero product property to find the zeros of each function.
xxxf 123)( 2 214)( 2 xxxf
0)4(3
0123 2
xx
xx
0
03
x
x
4
04
x
x
7
07
x
x
3
03
x
x
0)3)(7(
02142
xx
xx
Homework
• Page 296• 31-65 odd
• This is a skill that is best learned by repetition. You need to do all of these problems and maybe more to become really good at this.
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