Expand. 1. (x – 2)2
2. (y + 6)2
Factor each. 3. x2 – 6x + 9
4. x2 + 10x + 25
Solve each. 5. (x – 2)2 = 25
6. (x + 3)2 = -8
Algebra II
1
Completing the Square
Algebra II
¡  Graphing
¡  Factoring
(Fact)
¡  Square Roots (SQR)
Today we will learn a fourth
method...
¡  Completing the Square (CTS)
Algebra II
3
¡
Factoring is the easiest method to solve a
quadratic, however some trinomials don't factor. ¡
When you cannot factor a trinomial, you can
“complete the square” so that you can solve the
trinomial by using square roots. ¡
A perfect square trinomial is one that can be
factored into (a+b)2 or (a-b)2
§  a2 + 2ab + b2 = (a+b)2
§  a2 - 2ab + b2 = (a-b)2
Algebra II
4
1. x2 – 12x + c
3. x2 – 8x + c
5. x2 – 4x + c
x2 – 12x + 36 x2 – 8x + 16
x2 – 4x + 4
(x – 6)2
(x – 4)2
(x – 2)2
2. x2 + 16x + c
4. x2 + 10x + c
x2 + 16x + 64
x2 + 10x + 25
x2 + 9x + 81/4
(x + 8)2
(x + 5)2
(x + 9/2)2
Algebra II
6. x2 + 9x + c
What pattern are you seeing? The c must be (b/2)2
5
1. Put in the form ax2 + bx = c. 2. Divide all terms by a. 3. Add (b/2)2 to both sides. 4. Factor the left side. **(remember it is a perfect square)**
5. Solve by square roots.
Algebra II
6
1. x2 – 6x + 11 = 0
2. x2 + 8x = 9 x2 – 6x = -11
x2 + 8x + 16 = 9 + 16
x2 – 6x + 9 = -11 + 9
(x + 4)2 = 25
x2 – 6x + 9 = -2
√(x + 4)2 = √25
(x – 3)2 = -2
x + 4 = ± 5
√(x – 3)2 = √-2
x = - 4 ± 5
x – 3 = ± i√2
x = 3 ± i√2
x = -4 + 5
x = - 9
x=1
x = -4 – 5 Algebra II
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3. 2x2 – 12x + 14 = 0
4.
2x(x + 10) = 6
2x2 – 12x = -14
2x2 + 20x = 6
x2 – 6x = -7
x2 + 10x = 3
x2 – 6x + 9 = -7 + 9
x2 + 10x + 25 = 3 + 25
(x – 3)2 = 2
(x + 5)2 = 28
√(x – 3)2 = √2
√(x + 5)2 = √28
x – 3 = ± √2
x + 5 = ± 2√7
x = 3 ± √2
x = -5 ± 2√7
Algebra II
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5. 3x2 – 12x – 15 = 0
6. 3x2 + 15x – 9 = 0
3x2 – 12x = 15
3x2 + 15x = 9
x2 – 4x = 5
x2 + 5x = 3
x2 – 4x + 4 = 5 + 4
x2 + 5x + 25/4 = 3 + 25/4
(x – 2)2 = 9
(x + 5/2)2 = 37/4
√(x – 2)2 = √9
√(x + 5/2)2 = √37/4
x – 2 = ± 3
x + 5/2 = ± √37/2
x = 2 ± 3
x = -5/2 ± √37/2
x=5
Algebra II
x = -1
9
7.
2x(x + 7) = -10
8. 5x2 – 10x + 15 = 0
2x2 + 14x = -10
5x2 – 10x = -15
x2 + 7x = -5
x2 – 2x = -3
x2 + 7x + 49/4 = -5 + 49/4
x2 – 2x + 1 = -3 + 1
(x + 7/2)2 = 29/4
(x – 1)2 = -2
√(x + 7/2)2 = √29/4
√(x – 1)2 = √-2
x + 7/2 = ± √29/2
x – 1 = ± √-2
x = -7/2 ± √29/2
x = 1 ± i√2
Algebra II
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1.
x2 – 10x + 7 = 0
3.
3x(x + 4) = -15
2.
x2 – 15x + 64 = 100 Algebra II
4.
2x2 + 10x – 3 = 13 11
Method 1:
1.
Rewrite the right
side of the
equation using
completing the
square techniques
Algebra II
Method 2:
1.
Find the vertex
2.
Use the same “a”
as the original
equation
3.
Plug vertex and “a”
into vertex form
y = a(x – h)2 + k
12
1. y = x2 – 8x + 11
y = x2 – 8x + 16 + 11 – 16
y=
(x2
– 8x + 16) + (11 – 16) y = (x – 4)2 – 5
y = (x – 4)2 – 5 Algebra II
2. y = 2x2 + 16x + 9
Find the Vertex: x = -b/2a = -16/4 = -4
y = 2(-4)2 +16(-4) + 9 = -23
Vertex: (-4, -23)
Use the same a as in
original problem. y = a(x – h)2 + k
y = 2(x + 4)2 – 23
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3. y = -3x2 + 18x – 8 Find the Vertex: x = -b/2a = -18/-6 = 3
y = -3(3)2 + 18(3) – 8 = 19
4. y = x2 + 6x + 16
y = x2 + 6x + 9 + 16 – 9
y = (x2 + 6x + 9) + (16 – 9) Vertex: (3, 19)
y = (x + 3)2 + 7
Use the same a as in
original problem. y = (x + 3)2 + 7
y = a(x – h)2 + k
y = -3(x – 3)2 + 19
Algebra II
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5. y = 3x2 – 24x + 5
6. y = -2x2 – 8x + 17
Find the Vertex: Find the Vertex: x = -b/2a = 24/6 = 4
x = -b/2a = 8/-4 = -2
y = 3(4)2 – 24(4) + 5 = -43 y = -2(-2)2 – 8(-2) + 17 = 25
Vertex: (4, -43)
Vertex: (-2, 25)
Use the same a as in
original problem. Use the same a as in
original problem. y = a(x – h)2 + k
y = a(x – h)2 + k
y = 3(x – 4)2 – 43
y = -2(x + 2)2 + 25
Algebra II
15
A golfer hits a golf ball on the fairway with an initial
velocity of 80 feet per second. The height h (in feet) of
the golf ball t seconds after it is hit can be modeled by
the function h(t) = -16t2 + 80 t + 0.1.
a.
How high was ball when it was hit? b.
How high is the ball after 1 second?
c.
Find the maximum height of the ball. d.
How long does the ball take to hit the ground? Algebra II
16
An online music store sells about 4000 songs each day
when it charges $1 per song. For each 5 cent increase in
price, about 8- fewer songs per day are sold. a.  Write an equation for the monthly revenue of the
online store. b.  How many price changes would cause the store to
make no money? c.  How many changes in price would cause the store to
make a maximum revenue? d.  How much would each song be? e.  How many songs would they sell a month? f.  What is the maximum monthly revenue? Algebra II
17
The height y (in feet) of a baseball t seconds after it is hit
can be modeled by the function h(t) = -16t2 + 96t + 3. a.
How high was ball when it was hit? b.
How high is the ball after 2 seconds? c.
Find the maximum height of the ball. d.
How long does the ball take to hit the ground? Algebra II
18
Solve by completing the square: 1. 3x2 – 6x + 10 = -2
2. 2x2 – 14x = 10
Put in vertex form: 3. Y = 2x2 – 8x + 12
4. Y = -3x2 + 6x - 5 Algebra II
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