Transcript Lesson 3

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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Space Bar to display the answers.
D
-4/8 > 4/8
Lesson 9-3
Factoring Trinomials:
x2 + bx + c
Click the mouse button or press the
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Objectives
• Factor trinomials of the form x2 + bx + c
• Solve equations of the form x2 + bx + c = 0
Vocabulary
• none – means nothing.
Factoring x2 + bx + c
• To factor quadratic trinomials of the form x2 + bx + c,
find two integers, m and n, whose sum is equal to b
and whose product is equal to c.
• Then write x2 + bx + c using the pattern (x + m)(x + n)
• Symbols: x2 + bx + c = (x + m)(x + n)
when m + n = b and m  n = c
• Examples:
– x2 + 5x + 6 = (x + 2)(x + 3), since 2 + 3 = 5 and 2  3 = 6
– x2 + 7x + 10 = (x + 2)(x + 5), since 2 + 5 = 7 and 2  5 = 10
– x2 + 9x + 18 = (x + 3)(x + 6), since 3 + 6 = 9 and 3  6 = 18
Multiplication and Division PoE
Properties of Equality (PoE) are based on the concept
that as long as you do the same thing to both sides
of an equation, then you have not changed anything.
• Multiplication PoE
– For any numbers a, b, and c, if a = b, then ac = bc
– You can multiply both sides of an equation by the same
thing without changing the equation
• Division PoE
– For any numbers a, b, and c with c ≠ 0, if a = b, then a/c = b/c
– You can divide both sides of an equation by the same thing
without changing the equation
• Multiplication and division are reciprocal actions
Example 1
Factor
In this trinomial,
and
You need to find the two
numbers whose sum is 7 and whose product is 12. Make
an organized list of the factors of 12, and look for the pair
of factors whose sum is 7.
Factors of 12 Sum of Factors
1, 12
2, 6
3, 4
13
8
7
The correct factors are 3 and 4.
Write the pattern.
Answer:
and
Example 1 cont
Check You can check the result by multiplying the
two factors.
F
O
I
L
FOIL method
Simplify.
Example 2
Factor
In this trinomial,
and
This means
is
negative and mn is positive. So m and n must both be
negative. Therefore, make a list of the negative factors of
27, and look for the pair whose sum is –12.
Factors of 27 Sum of Factors
–1, –27
–3, –9
–28
–12
The correct factors are –3 and –9.
Write the pattern.
Answer:
and
Example 2 cont
Check You can check this result by using a graphing
calculator. Graph
and
on the same screen. Since only one graph appears,
the two graphs must coincide. Therefore, the trinomial
has been factored correctly.
Example 3
Factor
In this trinomial,
and
This means
is
positive and mn is negative, so either m or n is negative,
but not both. Therefore, make a list of the factors of –18
where one factor of each pair is negative. Look for the pair
of factors whose sum is 3.
Factors of –18
1, –18
–1, 18
2, –9
–2, 9
3, –6
–3, 6
Sum of Factors
–17
17
– 7
7
– 3
The correct factors are –3 and 6.
3
Write the pattern.
Answer:
and
Example 4
Factor
Since
and
is negative and mn is
negative. So either m or n is negative, but not both.
Factors of –20 Sum of Factors
1, –20
–1, 20
2, –10
–2, 10
4, –5
–4, 5
–19
19
– 8
8
– 1
1
The correct factors are 4 and –5.
Write the pattern.
Answer:
and
Example 5
Solve
Original equation
Rewrite the equation so that
one side equals 0.
Factor.
or
Zero Product Property
Solve each equation.
Answer: The solution is
Example 6
Architecture Marion has a small art studio measuring
10 feet by 12 feet in her backyard. She wants to build a
new studio that has three times the area of the old
studio by increasing the length and width by the same
amount. What will be the dimensions of the new
studio?
Explore Begin by
making a diagram like the
one shown to the right,
labeling the appropriate
dimensions.
Example 6 cont
Plan
Let
the amount added to each dimension of
the studio.
The new length times the new width equals the new area.
old area
Solve
Write the equation.
Multiply.
Subtract 360 from
each side.
Example 6 cont
Factor.
or
Zero Product
Property
Solve each equation.
Examine The solution set is
Only 8 is a valid
solution, since dimensions cannot be negative.
Answer: The length of the new studio should be
or 20 feet and the new width should be
or 18 feet.
Summary & Homework
• Summary:
– Factoring x2 + bx +c: Find m and n whose
sum is b and whose product is c.
– Then write x2 + bx + c as (x + m)(x + n)
• Homework:
– Pg. 493. 18-34 even, 38,40,48