Bell Ringer 2/20/15
Completely Factor & Check your answer.
1. Factor: 2x2 - 14x + 12
2. Factor: y2 + 4y + 4
3. Factor: 75x2 – 12
Objective
The student will be able to:
factor perfect square trinomials.
Factoring Chart
This chart will help you to determine
which method of factoring to use.
Type
Number of Terms
1. GCF
2. Grouping
3. Trinomials
2 or more
4
3
Review: Multiply (y + 2)2
(y + 2)(y + 2)
Check this out…whaaat!!
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
y2
Using the formula,
+2y (y + 2)2 = (y)2 + 2(y)(2) + (2)2
2 = y2 + 4y + 4
(y
+
2)
+2y
+4
Which one is quicker?
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
y2 + 4y + 4
1) Factor x2 + 6x + 9
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = x
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(x + 3)2
Yes, b = 3
3) Is the middle term twice the
You can still
product of the a and b?
factor the other way
but this is quicker!
Yes, 2ab = 2(x)(3) = 6x
2) Factor y2 – 16y + 64
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = y
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(y – 8)2
Yes, b = 8
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(y)(8) = 16y
Factor m2 – 12m + 36
1.
2.
3.
4.
(m – 6)(m + 6)
(m – 6)2
(m + 6)2
(m – 18)2
3) Factor 4p2 + 4p + 1
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = 2p
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(2p + 1)2
Yes, b = 1
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(2p)(1) = 4p
Objective
The student will be able to:
factor using difference of squares.
Factoring Chart
This chart will help you to determine
which method of factoring to use.
Type
Number of Terms
1. GCF
2. Grouping
3. Trinomials
4. Difference of Squares
2 or more
4
3
2
Determine the pattern
1
4
9
16
25
36
…
= 12
= 22
= 32
= 42
= 52
= 62
These are perfect squares!
You should be able to list
the first 15 perfect
squares …
Perfect squares
1, 4, 9, 16, 25, 36, 49, 64, 81,
100, 121, 144, 169, 196, 225
Review: Multiply (x – 2)(x + 2)
First terms: x2
Outer terms: +2x
Inner terms: -2x
Last terms: -4
Combine like terms.
x2 – 4
Notice the
middle terms
eliminate
each other!
x
-2
x2
-2x
+2 +2x
-4
x
This is called the difference of squares.
Difference of Squares
2
2
a - b = (a - b)(a + b)
or
2
2
a - b = (a + b)(a - b)
The order does not matter!!
4 Steps for factoring
Difference of Squares
1. Are there only 2 terms?
2. Is the first term a perfect square?
3. Is the last term a perfect square?
4. Is there subtraction (difference) in the
problem?
If all of these are true, you can factor
using this method!!!
1. Factor x2 - 25
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
x2 – 25
Two terms? Yes
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
( x + 5 )(x - 5 )
Write your answer!
2. Factor 16x2 - 9
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
16x2 – 9
Two terms? Yes
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
(4x + 3 )(4x - 3 )
Write your answer!
3. Factor 81a2 – 49b2
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
81a2 – 49b2
Two terms? Yes
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
(9a + 7b)(9a - 7b)
Write your answer!
Factor
1.
2.
3.
4.
2
x
(x + y)(x + y)
(x – y)(x + y)
(x + y)(x – y)
(x – y)(x – y)
Remember, the order doesn’t matter!
–
2
y
Factor
1.
2.
3.
4.
2
18c
prime
2(9c2 + 4d2)
2(3c – 2d)(3c + 2d)
2(3c + 2d)(3c + 2d)
You cannot factor using
difference of squares
because there is no
subtraction!
+
2
8d
Objective
The student will be able to:
use the zero product property to
solve equations
Zero Product Property
If a • b = 0 then
a=0,
b=0,
or both a and b equal 0.
4 steps for solving a quadratic equation
1. Set the equation equal to 0.
2. Factor the equation.
3. Set each part equal to 0 and
solve.
4. Check your answer on the
calculator if available.
Set = 0
Factor
Split/Solve
Check
1. Solve (x + 3)(x - 5) = 0
Using the Zero Product Property,
you know that either
x + 3 = 0 or x - 5 = 0
Solve each equation.
x = -3 or x = 5
{-3, 5}
2. Solve (2a + 4)(a + 7) = 0
2a + 4 = 0 or a + 7 = 0
2a = -4 or a = -7
a = -2 or a = -7
{-2, -7}
3. Solve (3t + 5)(t - 3) = 0
3t + 5 = 0 or t - 3 = 0
3t = -5 or t = 3
t = -5/3 or t = 3
{-5/3, 3}
4. Solve
2
x
- 11x = 0
GCF = x
x(x - 11) = 0
x = 0 or x - 11 = 0
x = 0 or x = 11
{0, 11}
Set = 0
Factor
Split/Solve
Check