¡  Complete
3.1 Exploration from
Student Journal Pg. 45 (use your calculator)
Answers: a. E b. D
Algebra II
c. C
d. A e. F
f. B
1
Solving Quadratics by
graphing, factoring, and
square roots
Algebra II
¡  Quadratic
variable: equation in one
A quadratic (degree two) that can be written
in the form ax2 + bx + c = 0
¡  Root
of an equation: A solution to the equation ax2 + bx + c = 0
¡  Zero
of a function: A solution to the equation ax2 + bx + c = 0, a
value where f(x) = 0. Algebra II
3
¡  By
graphing
Graph and find the x-intercepts
¡  Using
Square Roots
Isolate the quadratic (squared) term. There can not be a linear (x to the first) term. ¡  By
factoring
Factor and solve using zero-product
property. Algebra II
4
1. 0 = x2 + 2x – 8
AOS: x = - b / (2a)
x = -2 / (2Ÿ1)
x = -1
y = (-1)2 + 2(-1) – 8
Vertex: (-1, -9)
Table à
Reflect
Algebra II
x
y
0
-8
1
-5
Solutions to
equation: (2, 0); (-4, 0)
5
2. x2 – 4x + 3 = 0
AOS: x = - b / (2a)
x = 4 / (2Ÿ1)
x = 2
y = (2)2 – 4(2) + 3
Vertex: (2, -1)
Table à
Reflect
Algebra II
X
Y
3
0
4
3
Solutions to
equation: (3, 0); (1, 0)
6
3. 2x = -x2 + 3
0 = -x2 – 2x + 3 AOS: x = - b / (2a)
x = 2 / (2Ÿ-1)
x = -1
y = -(-1)2 – 2(-1) + 3
Vertex: (-1, 4)
Table à
Reflect
Algebra II
X
Y
0
3
1
0
Solutions to
equation: (-3, 0); (1, 0)
7
Isolate the squared term
2.  Take the square root of both
sides
1. 
­  if you introduce a √ into the
problem, then YOU MUST put a
± into the answer
3. 
Simplify, if possible
Algebra II
8
1. 2x2 + 1 = 17
2x2 = 16
2. ⅓(x + 5)2 = 7
3(⅓)(x + 5)2 = 7 ∙ 3
x2 = 8
(x + 5)2 = 21
√x2 = √8
√(x + 5)2 = √21
x = ± √4 ∙ √2
x + 5 = ± √21
x = ± 2√2
Algebra II
x = –5 ± √21 9
3. 4x2 – 6 = 42
4. 2(x – 4)2 = 50 4x2 = 48
(x – 4)2 = 25
x2 = 12
√(x – 4)2 = √25
√x2 = √12
x – 4 = ± 5
x = ± 2√3
x = 4 ± 5
x = 4 + 5 x = 4 – 5
x=9
Algebra II
x = -1
10
5. 6 – p2 = - 4
8
6. ½(2x – 3)2 = 18
- p2 = - 10
8
(2x – 3)2 = 36
√(2x – 3)2 = √36
2x – 3 = ± 6
- p2 = - 80
p2
= 80
2x = 3 ± 6
2x = 9
√p2 = √80
p = ± 4√5
Algebra II
x = 9/2
2x = - 3
x = -3/2
11
If a · b = 0, what do you
know about a or b?
Algebra II
12
If a and b are real numbers and
ab = 0, then a = 0 or b = 0.
If the product of two numbers is zero, then
one of the factors must be zero.
Algebra II
13
Step 1: Write the equation in standard form.
Step 2: Factor completely.
Step 3: Set each factor equal to 0.
Step 4: Solve the resulting equations.
Step 5: C
heck each solution in the original
equation.
Algebra II
14
1. x2 + 3x – 18 = 0
2. 2p2 - 17p + 45 = 3p – 5
(x + 6)(x – 3) = 0
2p2 – 20p + 50 = 0
x + 6 = 0 x – 3 = 0
2(p2 – 10p + 25) = 0
x=–6
x = 3
2(p – 5)(p – 5) = 0
2 = 0 p – 5 = 0 p – 5 = 0
p=5
Algebra II
p = 5
15
3. 8x2 – 3x + 2 = 7
4. 3x – 6 = x2 – 10
8x2 – 3x – 5 = 0
0 = x2 – 3x – 4
(8x + 5)(x – 1) = 0
0 = (x – 4)(x + 1)
8x + 5 = 0 x – 1 = 0
x – 4 = 0 x + 1 = 0
x = – 5/8
x = 1
x=4
x = –1 Algebra II
16
5. 2w2 - 10w = 23w – w2
6. 4x2 = 16
3w2 – 33w = 0
4x2 – 16 = 0
3w(w – 11) = 0
4(x2 – 4) = 0
3w = 0 w – 11 = 0
4(x – 2)( x + 2) = 0
w=0
w = 11
4 = 0 x - 2 = 0 x + 2 = 0
x=2
Algebra II
x=-2
17
7. 6x2 – 17x = 3
8. 32x3 – 50x = 0
6x2 – 17x - 3 = 0
2x(16x2 – 25) = 0
(6x + 1)(x – 3) = 0
2x(4x – 5)(4x + 5) = 0
6x + 1 = 0 x – 3 = 0
2x = 0 4x – 5 = 0 4x + 5 = 0
x = - 1/6
x = 3
x=0
x = 5/4
x = -5/4
Algebra II
18
Algebra II
19
¡  Intercept
¡  f(x)
form = factored form
= a(x – p)(x – q)
¡  solutions
= roots = à
= #
x
= zeros = x-intercepts à
( #, 0)
same number, just means different things
Algebra II
20
1. f(x) = 3x2 + 14x – 5
2. f(x) = x2 – 8x + 15
f(x) = (3x – 1)(x + 5)
f(x) = (x – 3)(x – 5)
3x – 1 = 0 x + 5 = 0
x – 3 = 0 x – 5 = 0
x = 1/3
x=–5
x=3
x = 5
(1/3, 0)
( – 5, 0) (3, 0)
(5, 0)
Algebra II
21
3. f(x) = 2x2 – 9x + 10 4. f(x)= x2 – 3x – 28
f(x) = (2x – 5)(x – 2)
f(x) = (x – 7)(x + 4)
2x – 5= 0 x – 2 = 0
x – 7 = 0 x + 4 = 0
x = - 5/2
x=2
x=7
x = -4
(-5/2, 0)
(2, 0) (7, 0)
(-4, 0)
Algebra II
22
1. 
4(x – 3)2 = 20
3. 
2x2 + 14 = 70
2. 
3x2 – 5x = 12 4. 
4x2 – 10 = 3x Algebra II
23
h(t) =
¡  t
2
-16t
+ h0
à the time in seconds
¡  h0
à the initial height
¡  h(t)
à the ending height
Algebra II
24
A stunt man working on the set of a
movie is to fall out of a window 100
feet above the ground. For the stunt
man's safety, an air cushion 25 feet
wide by 30 feet long by 9 feet high is
positioned on the ground below the
window. A. 
For how many seconds will the
stunt man fall before he reaches
the cushion?
B. 
A movie camera operating at a
speed of 24 frames per second
records the stunt man's fall. How
many frames of film show the
stunt man falling? Algebra II
25
The tallest building in the United
States is the Sears Tower in Chicago,
Illinois. It is 1450 feet tall. A. 
How long would it take a penny
to drop from the top of this
building? B. 
How fast would the penny be
traveling when it hits the ground
if the speed is given by s = 32t
where t is the number of seconds
since the penny was dropped? Algebra II
26
You have made a rectangular stained glass window
that is 2 feet by 4 feet. You have 7 square feet of
clear glass to create a uniform border. What should
the width of the border be? Algebra II
27
You are putting a uniform border around your
rectangular garden that is 10 by 12 feet. You have
enough brick to cover 48 square feet. What is the
width of the border? Algebra II
28
You maintain a music-oriented web-site that allows
subscribers to download audio and video clips.
When the subscription price is $16 per year, you get
30,000 subscribers. For each $1 increase in price,
you expect to lose 1,000 subscribers. How much
should you charge to maximize your annual
revenue? What is your maximum revenue? Algebra II
29
The owner of a gym charges $34 per month and has
48 members. For every $1 decrease in price they
would gain four new members. What should the
gym charge to maximize your annual revenue, and
what is the maximum revenue? Algebra II
30
The length of a rectangle is three more than
twice the width. Determine the dimensions
that will give a total area of 27 m2. Algebra II
31
The length of a Ping-Pong table is 3 ft more than
twice the width. The area of a Ping-Pong table is 90
square feet. What are the dimensions of a PingPong table?
Algebra II
32
Find the value of x, given the
area of the triangle is 24. X+2
2X – 6 Algebra II
33
Solve the quadratic by any method:
1. 4x2 = 20
2. 3x2 – 5x – 18 = x2 – 2x + 2
3. 4(x – 3)2 = 16
4. Write in intercept form and find the zeros.
f(x) = 4x2 – 12x – 16 5. How long will it take an object dropped
from a 550 foot tall tower to land on the roof
of a 233 foot tall building?
Algebra II
34