6.4 Completing the Square
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6.4 Completing the Square
The Square Root Property
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What does c equal to make the quadratic equation a prefect square
x 2 + 16x + c ; ( x + )2
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Square Root Property
Solve x2 – 16 = 0
Factor into (x + 4)(x – 4) = 0
So the answers are 4, -4
This can be solve another way
There are always two
answers. 4
16
162
x
x
x
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Solve Using factoring and Square Root property
x2 + 16x + 64 = 36
(x + 8)(x + 8) = 36
(x + 8)2 = 36
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Solve Using factoring and Square Root property
(x + 8)2 = 36
14 ,2
68
68
368
x
x
x
x
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Solve x2 – 10x + 25 = 12
Factor
125
125
2
2
x
x
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Solve x2 – 10x + 25 = 12
Factor
325
325
125
125
2
2
x
x
x
x
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Solve x2 – 10x + 25 = 12
Factor
325
325
125
125
2
2
x
x
x
x
325
325
x
x
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Complete the Square
In case the quadratic equation is not a prefect square, we can force this to happen.
If a = 1, then we move c to the other side and add half of b squared to both side of the equation.
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Complete the Square
If a = 1, then we move c to the other side and add half of b squared to both side of the equation.
22
222
2
2
22
22
0
bc
bx
bc
bbxx
cbxx
cbxx
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Solve by Completing the Square
162
2122
44
124
0124
2
22
2
2
2
x
xx
xx
xx
6,2;42
42
xx
x
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If a ≠ 1
abyDivide
cbxax
cbxax
2
2 0
22
222
2
22
22
a
b
a
c
a
bx
a
b
a
c
a
bx
a
bx
a
cx
a
bx
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Solve 3x2 -2x – 1 = 0
3
1
3
2
123
0123
2
2
2
xx
xx
xx
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Solve 3x2 -2x – 1 = 0
3
1
3
2
123
0123
2
2
2
xx
xx
xx
9
4
9
1
9
3
3
1
3
1
3
1
3
2
2
1
3
2
2
222
x
xx
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Solve 3x2 -2x – 1 = 0
3
1
3
2
123
0123
2
2
2
xx
xx
xx
3
2
9
4
3
1
9
4
9
1
9
3
3
1
3
1
3
1
3
2
2
1
3
2
2
222
x
x
xx
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Solve 3x2 -2x – 1 = 0
3
1
3
2
123
0123
2
2
2
xx
xx
xx
3
1 ;1
3
2
3
1
3
2
9
4
3
1
9
4
9
1
9
3
3
1
3
1
3
1
3
2
2
1
3
2
2
222
xx
x
x
x
xx
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Solve x2 + 2x + 3 = 0
Solve by completing the square.
132
22
32
032
22
2
2
xx
xx
xx
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Solve x2 + 2x + 3 = 0
Solve by completing the square.
132
22
32
032
22
2
2
xx
xx
xx
21
21 2
x
x
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Solve x2 + 2x + 3 = 0
Solve by completing the square.
132
22
32
032
22
2
2
xx
xx
xx
21
21 2
x
x
21
21
ix
ix
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Homework
Page 310 – 311
# 15 – 17 odd
25 – 47 odd
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Homework
Page 310 – 311
# 14 – 18 even
24 – 40 even