Transcript Lesson 6-2

5-Minute Check on Chapter 2
Transparency 3-1
1. Evaluate 42 - |x - 7| if x = -3
2. Find 4.1  (-0.5)
Simplify each expression
4. (36d – 18) / (-9)
3. 8(-2c + 5) + 9c
5. A bag of lollipops has 10 red, 15 green, and 15 yellow lollipops.
If one is chosen at random, what is the probability that it is
not green?
6.
Standardized Test Practice:
Which of the following is a true
statement
A
8/4 < 4/8
B
-4/8 < -8/4
C
-4/8 > -8/4
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D
-4/8 > 4/8
Lesson 9-6
Perfect Squares and Factoring
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Objectives
• Factor perfect square trinomials
• Solve equations involving perfect squares
Vocabulary
• Perfect square- a number whose square root is a
rational number
• trinomial– the sum of three monomials
Factoring Perfect Square Trinomials
• If a trinomial can be written in the form:
a2 + 2ab + b2 or a2 – 2ab + b2
then it can be factored as
(a + b)2 or as (a – b)2 respectively
• Symbols:
a2 + 2ab + b2 = (a + b)2 or a2 – 2ab + b2 = (a – b)2
• Examples:
x2 + 6x + 9 = (x + 3)2
4x2 – 24x + 36 = (2x – 6)2
Example 1a
Determine whether
trinomial. If so, factor it.
is a perfect square
1. Is the first term a perfect square?
Yes,
2. Is the last term a perfect square?
Yes,
3. Is the middle term equal to
Answer:
? Yes,
is a perfect square trinomial.
Write as
Factor using the pattern.
Example 1b
Determine whether
square trinomial. If so, factor it.
is a perfect
1. Is the first term a perfect square?
Yes,
2. Is the last term a perfect square?
Yes,
3. Is the middle term equal to
Answer:
? No,
is not a perfect square trinomial.
Example 2a
Factor
.
First check for a GCF. Then, since the polynomial has two
terms, check for the difference of squares.
6 is the GCF.
and
Answer:
Factor the difference
of squares.
Example 2b
Factor
.
This polynomial has three terms that have a GCF of 1.
While the first term is a perfect square,
the last term is not. Therefore, this is not a perfect
square trinomial.
This trinomial is in the form
Are there two
numbers m and n whose product is
and whose sum is 8? Yes, the product of 20 and –12 is
–240 and their sum is 8.
Example 2b cont
Write the pattern
and
Group terms with
common factors
Factor out the GCF
from each grouping
Answer:
is the
common factor.
Example 3
Solve
Original equation
Recognize
as a perfect square trinomial.
Factor the perfect square
trinomial.
Set the repeated factor
equal to zero.
Solve for x.
Answer: Thus, the solution set is
Check this
solution in the original equation.
Example 4a
Solve
.
Original equation
Square Root Property
Add 7 to each side.
or
Separate into two equations.
Simplify.
Answer: The solution set is
Check each
solution in the original equation.
Example 4b
Solve
.
Original equation
Recognize perfect square
trinomial
Factor perfect square trinomial
Square Root Property
Subtract 6 from each side.
or
Separate into two equations.
Simplify.
Answer: The solution set is
Example 4c
Solve
.
Original equation
Square Root Property
Subtract 9 from each side.
Answer: Since 8 is not a perfect square, the solution set is
Using a calculator, the approximate
solutions are
or about –6.17 and
or about –11.83.
Example 4c cont
Check You can check your answer using a graphing
calculator. Graph
and
Using the
INTERSECT feature of your graphing calculator, find
where
The check of –6.17 as one of the
approximate solutions is shown.
Factoring Techniques
• Greatest Common Factor (GCF)
• Factor out a Common Factor
• Difference of Squares
• Perfect Square Trinomials
• Factoring by Grouping
Summary & Homework
• Summary:
– If a trinomial can be written in the form
a2 + 2ab + b2 or a2 – 2ab + b2,
then it can be factored as (a + b)2 or (a – b)2, respectively
– For a trinomial to be factorable as a perfect square, the first
term must be a perfect square, the middle must be twice the
product of the square roots of the first and last terms, and
the last term must be a perfect square
– Square Root Property:
for any number n>0, if x2 = n, then x = +- √n
• Homework:
– Pg. 512 18-22,26-38,44,46