1 completing the square 2 what do you get when you foil the following expressions? (x + 1) (x+1)= (x...
TRANSCRIPT
1
Completing the Square
Completing the Square
2
What do you get when you foil the following expressions?
(x + 1) (x+1)=
(x + 2) (x+2) =
(x + 3) (x+3) =
(x + 4) (x+4) =
(x + 5) (x+5) =
(x + 6)2 =
(x + 7)2 =
(x + 8)2 =
(x + 9)2 =
(x + 10)2 =
3
What do you get when you foil the following expressions?
(x + 1)2 = x2 + 2x + 1
(x + 2)2 = x2 + 4x + 4
(x - 3)2 = x2 - 6x + 9
(x - 4)2 = x2 - 8x + 16
(x + 5)2 = x2 + 10x + 25
(x + 10)2 = x2 + 20x + 100
(x - 13)2 = x2 - 26x + 169
(x - 25)2 = x2 - 50x + 625
(x – 0.5)2 = x2 - x + 0.25
(x – 3.2)2 = x2 – 6.4x + 10.24
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Fill in the missing number to complete a perfect square.
x2 + 2x + ____
x2 + 8x + ___
x2 + 6x + ___
x2 - 14x + ___
x2 – 20x + ___
x2 + 16x + _____
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Fill in the missing number to complete a perfect square.
x2 + 10x + ___x2 + 10x + 25 = (x + 5)2
x2 + 18x + ___x2 + 18x + 81 = (x + 9)2
x2 + 12x + ___x2 + 12x + 36 = (x + 6)2
x2 - 30x + ___x2 - 30x + 225 = (x - 15)2
x2 – 2.8x + ___x2 – 2.8x + 1.96 = (x – 1.4)2
x2 + 0.5x + _____x2 + 0.5x + 0.0625 = (x – 0.25)2
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Changing from standard form to vertex form
By completing the square on a quadratic in standard form, it is changed into vertex form
Change to vertex form:
y = x2 + 14x - 10
y = x2 + 14x + ____ - 10
y = x2 + 14x + 49 - 10 - 49
y = (x + 7)2 -59
The vertex is at (-7, -59)
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Changing from standard form to vertex form
By completing the square on a quadratic in standard form, it is changed into vertex form
Change to vertex form:
y = x2 - 12x + 5
y = x2 - 12x + ____ + 5
y = x2 - 12x + 36 + 5 - 36
y = (x - 6)2 - 31
The vertex is at (6, -31)
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Changing from standard form to vertex form
By completing the square on a quadratic in standard form, it is changed into vertex form
Change to vertex form:
y = x2 - 28x + 200
y = x2 - 28x + ____ + 200
y = x2 - 28x + 196 + 200 - 196
y = (x - 14)2 + 4
The vertex is at (14, 4)
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Changing from standard form to vertex form
By completing the square on a quadratic in standard form, it is changed into vertex form
Change to vertex form:
y = x2 – 0.75x - 1
y = x2 – 0.75x + ____ + - 1
y = x2 – 0.75x + .140625 - 1 - .140625
y = (x – 0.375)2 – 1.140625
The vertex is at (0.375, -1.140625)
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Change to vertex form:
y = x2 + 4x + 10
y = x2 + 4x + ___ + 10
y = x2 + 4x + 4 + 10 - 4
y = (x + 2)2 + 6
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Change to vertex form:
y = x2 + 19x - 1
y = x2 + 19x + ___ - 1
y = x2 + 19x + 90.25 - 1 – 90.25
y = (x + 9.5)2 - 91.25
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More Complicated Versions of Completing the Square
If the leading coefficient is not equal to 1, completing the square is slightly more difficult.
Directions for Completing the Square:
1.) Move the constant out of the way.
2.) Factor out A from the x2 and x term.
3.) Determine what is half of the remaining B.
4.) Square it and put this in for C.
5.) Put in a constant to cancel out the last step.
6.) Write the parenthesis as a perfect square and simplify everything else.
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Change to vertex form:
y = 2x2 + 4x + 10
y = 2(x2 + 2x + ___) + 10 - ___
y = 2(x2 + 2x + 1) + 10 - 2
y = 2(x + 1)2 + 8
Vertex at (-1, 8)
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Change to vertex form:
y = 3x2 + 12x + 22
y = 3(x2 + 4x + ___) + 22 - ___
y = 3(x2 + 4x + 4) + 22 - 12
y = 3(x + 2)2 + 10
Vertex at (-2, 10)
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Change to vertex form:
y = 6x2 - 48x + 65
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Change to vertex form:
y = 7x2 - 98x + 400
17
Change to vertex form:
y = 12x2 - 60x + 312
18
Change to vertex form:
y = -5x2 + 20x - 32
y = -5(x2 - 4x + ___) - 32 - ___
y = -5(x2 - 4x + 4) - 32 + 20
y = -5(x - 2)2 - 12
Vertex at (2, -12)
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Change to vertex form:
y = -6x2 + 72x - 53
y = -6(x2 - 12x + ___) - 53 - ___
y = -6(x2 - 12x + 36) - 53 + 216
y = -6(x - 6)2 + 163
Vertex at (6, 163)
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Methods of Locating the Vertex of a Parabola:
If the quadratic is in vertex form:
𝑦=𝑎 (𝑥−h )2+𝑘 The vertex is @ (h, k):
If the quadratic is in factored form:
𝑦=𝑎 (𝑥−¿) (𝑥−¿)The x value of the vertex is halfway between the roots. Plug in & solve to find the y value.
If the quadratic is in standard form:
𝑦=𝑎𝑥2+𝑏𝑥+𝑐Complete the square to change to vertex form.
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Change to vertex form:
y 5x 2 3x 2
y 5 x 2 3
5x ___
2 ___
y 5 x 2 3
5x
9
100
2
45
100
y 5 x 3
10
2
245
100
Vertex at (-0.3, -2.45)
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Change to vertex form:
y 7x 2 9x 25
y 7 x 2 9
7x ___
25 ___
y 7 x 2 9
7x
81
196
25
81
28
y 7 x 9
14
2
619
28
28
619,
14
9
23
Change to vertex form:
xxy 85 2
24
Change to vertex form:
322
1 2 xxy
25
Solve by completing the square.
542 xx
542 xx
__5__42 xx
45442 xx
92 2 x
92 2 x
92 2 x
32 x
23 x
1,5 x
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Solve by completing the square.
xx 1252 2
5122 2 xx
__5__62 2 xx
185962 2 xx
2332 2 x
2
233 2 x
2
233 2 x
2
233 2 x
2
233 x
2
233 x
391.0,391.6 x
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x2 + 6x - 8 = 0
x2 + 6x = 8
x2 + 6x + ___= 8 + ___
x2 + 6x + 9 = 8 + 9
(x+3)2 = 17
17 3 x
173x
Example: Solve by completing the square: x2 + 6x – 8 = 0
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Solve by completing the square:
cbxax 20
cbxax 2
____2
cx
a
bxa
a
bc
a
bx
a
bxa
44
2
2
22
ca
b
a
bxa
42
22
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Solve by completing the square:
ca
b
a
bxa
42
22
a
c
a
b
a
bx
2
22
42
a
c
a
b
a
bx
2
22
42
a
c
a
b
a
bx
2
2
42
a
c
a
b
a
bx
2
2
42
22
2
4
4
42 a
ac
a
b
a
bx
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Solve by completing the square:
22
2
4
4
42 a
ac
a
b
a
bx
2
2
4
4
2 a
acb
a
bx
a
acb
a
bx
2
4
2
2
a
acbbx
2
42
This is called the Quadratic Formula.
You must memorize it!!!