Transcript lalg1_fl_ch09_06

9.6 Factor ax2 + bx + c
Warm Up
Lesson Presentation
Lesson Quiz
9.6
Warm-Up
Find the product.
1. (3c + 3)(2c – 3)
ANSWER
6c2 – 3c – 9
2. (2y + 3)(2y + 1)
ANSWER
4y2 + 8y + 3
3. A cat leaps into the air with an initial velocity of
12 feet per second to catch a speck of dust, and then
falls back to the floor. How long does the cat remain
in the air?
ANSWER
0.75 sec
9.6
Example 1
Factor 2x2 – 7x + 3.
SOLUTION
Because b is negative and c is positive, both factors
of c must be negative. Make a table to organize your
work.
You must consider the order of the factors of 3,
because the x-terms of the possible factorizations
are different.
9.6
Example 1
Correct
ANSWER 2x2 – 7x + 3 = (x – 3)(2x – 1)
9.6
Example 2
Factor 3n2 + 14n – 5.
SOLUTION
Because b is positive and c is negative, the factors
of c have different signs.
9.6
Example 2
Correct
ANSWER 3n2 + 14n – 5 = (n + 5)(3n – 1)
9.6
Guided Practice
Factor the trinomial.
1. 3t2 + 8t + 4
ANSWER
(t + 2)(3t + 2)
2. 4s2 – 9s + 5
ANSWER
(s – 1)(4s – 5)
ANSWER
(h + 7)(2h – 1)
3.
2h2 + 13h – 7
9.6
Example 3
Factor –4x2 + 12x + 7.
SOLUTION
STEP 1
Factor –1 from each term of the trinomial.
–4x2 + 12x + 7 = –(4x2 – 12x – 7)
STEP 2
Factor the trinomial 4x2 – 12x – 7. Because b and c are
both negative, the factors of c must have different
signs. As in the previous examples, use a table to
organize information about the factors of a and c.
9.6
Example 3
Correct
ANSWER
–4x2 + 12x + 7 = –(2x + 1)(2x – 7)
9.6
Example 3
CHECK
You can check your factorization using a
graphing calculator. Graph y1 = –4x2 + 12x + 7
and y2 = (2x + 1)(2x – 7). Because the graphs
coincide, you know that your factorization is
correct.
9.6
Guided Practice
Factor the trinomial.
4. –2y2 – 5y – 3
ANSWER
–(y + 1)(2y + 3)
5.
–5m2 + 6m – 1
ANSWER
–(m – 1)(5m – 1)
6.
–3x2 – x + 2
ANSWER
–(x + 1)(3x – 2)
9.6
Example 4
DISCUS
An athlete throws a
discus from an initial
height of 6 feet and
with an initial vertical
velocity of 46 feet per
second.
a. Write an equation that gives the height (in feet) of
the discus as a function of the time (in seconds)
since it left the athlete’s hand.
b. After how many seconds does the discus hit the
ground?
9.6
Example 4
SOLUTION
a.
b.
Use the vertical motion model to write an
equation for the height h (in feet) of the discus. In
this case, v = 46 and s = 6.
h = –16t2 + vt + s
Vertical motion model
h = –16t2 + 46t + 6
Substitute 46 for v and 6 for s.
To find the number of seconds that pass before
the discus lands, find the value of t for which the
height of the discus is 0. Substitute 0 for h and
solve the equation for t.
9.6
Example 4
0 = –16t2 + 46t + 6
Substitute 0 for h.
0 = –2(8t2 – 23t – 3)
Factor out –2.
0 = –2(8t + 1)(t – 3)
Factor the trinomial. Find
factors of 8 and –3 that produce
a middle term with a coefficient
of –23.
8t + 1 = 0 or t – 3 = 0
t = – 1 or
8
t=3
Zero-product property
Solve for t.
9.6
Example 4
The solutions of the equation are – 81 and 3. A negative
solution does not make sense in this situation, so
disregard – 1 .
8
ANSWER
The discus hits the ground after 3 seconds.
9.6
7.
Guided Practice
WHAT IF? In Example 4, suppose another athlete
throws the discus with an initial vertical velocity of
38 feet per second and releases it from a height of 5
feet. After how many seconds does the discus hit
the ground?
ANSWER
The discus hits the ground after 2.5 seconds.
9.6
8.
Guided Practice
SHOT PUT In a shot put event, an athlete throws
the shot put from an initial height of 6 feet and with
an initial vertical velocity of 29 feet per second. After
how many seconds does the shot put hit the
ground?
ANSWER
The shot put hits the ground after 2 seconds.
9.6
Example 5
w(3w + 13) = 10 Write an equation to model area.
3w2 + 13w2 – 10 = 0 Simplify and subtract 10 from each side.
(w + 5)(3w – 2) = 0 Factor left side.
w + 5 = 0 or 3w – 2 = 0 Zero-product property
2
w
w = – 5 or
= 3 Solve for w.
Reject the negative width.
ANSWER
The correct answer is A.
9.6
9.
Guided Practice
A rectangle’s length is 1 inch more than twice its
width. The area is 6 square inches. What is the width?
A
1m
2
ANSWER
3m
B 2
B
C 2m
D
3 m
2
9.6
Lesson Quiz
Factor the trinomial.
1. – x2 + x + 30
ANSWER
2.
5b2 +3b – 14
ANSWER
3.
(b + 2)(5b – 7)
6y2 – 13y – 5
ANSWER
4.
– (x + 5)(x – 6)
(3y + 1)(2y – 5)
Solve 2x2 + 7x = – 3
ANSWER
– 1 , –3
2
9.6
5.
Lesson Quiz
A baseball is hit into the air at an initial height of
4 feet and an initial velocity of 30 feet per second.
For how many seconds is it in the air?
ANSWER
2 sec