Transcript Completing the Square presentation
Completing the Square
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What do you get when you foil the following expressions?
(x + 1) (x+1)= (x + 6) 2 = (x + 2) (x+2) = (x + 7) 2 = (x + 3) (x+3) = (x + 8) 2 = (x + 4) (x+4) = (x + 9) 2 = (x + 5) (x+5) = (x + 10) 2 = 2
What do you get when you foil the following expressions?
(x + 1) 2 =
x 2 + 2x + 1
(x + 10) 2 =
x 2 + 20x + 100
(x + 2) 2 =
x 2 + 4x + 4
(x - 13) 2 =
x 2 - 26x + 169
(x - 3) 2 =
x 2 - 6x + 9
(x - 25) 2 =
x 2 - 50x + 625
(x - 4) 2 =
x 2 - 8x + 16
(x + 5) 2 =
x 2 + 10x + 25
(x – 0.5) 2 =
x 2 - x + 0.25
(x – 3.2) 2 =
x 2 – 6.4x + 10.24
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Fill in the missing number to complete a perfect square.
x 2 + 2x + ____ x 2 - 14x + ___ x 2 + 8x + ___ x 2 – 20x + ___ x 2 + 6x + ___ x 2 + 16x + _____ 4
x 2 x 2
Fill in the missing number to complete a perfect square.
= (x + 5) 2 x 2 = (x - 15) 2 = (x + 9) 2 x 2 = (x – 1.4) 2 x 2 = (x + 6) 2 x 2 = (x – 0.25) 2 5
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 = x 2 + 14x - 10 y = x 2 + 14x + ____ - 10 y = x 2 + 14x + 49 - 10 49 y = (x + 7) 2 -59 The vertex is at (-7, -59) 6
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 = x 2 - 12x + 5 y = x 2 - 12x + ____ + 5 y = x 2 - 12x + 36 + 5 36 y = (x - 6) 2 - 31 The vertex is at (6, -31) 7
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 = x 2 - 28x + 200 y = x 2 - 28x + ____ + 200 y = x 2 - 28x + 196 + 200 196 y = (x - 14) 2 + 4 The vertex is at (14, 4) 8
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 = x 2 – 0.75x - 1 y = x 2 – 0.75x + ____ + - 1 y = x 2 – 0.75x + .140625
- 1 .140625
y = (x – 0.375) 2 – 1.140625
The vertex is at (0.375, -1.140625) 9
Change to vertex form:
y = x 2 + 4x + 10 y = x 2 + 4x + ___ + 10 y = x 2 + 4x + 4 + 10 - 4 y = (x + 2) 2 + 6 10
Change to vertex form:
y = x 2 + 19x - 1 y = x 2 + 19x + ___ - 1 y = x 2 + 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 x 2 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 = 2x 2 + 4x + 10 y = 2(x 2 + 2x + ___) + 10 - ___ y = 2(x 2 + 2x + 1 ) + 10 2 y = 2(x + 1) 2 + 8
Vertex at (-1, 8)
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Change to vertex form:
y = 3x 2 + 12x + 22 y = 3(x 2 + 4x + ___) + 22 - ___ y = 3(x 2 + 4x + 4 ) + 22 12 y = 3(x + 2) 2 + 10
Vertex at (-2, 10)
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Change to vertex form:
y = 6x 2 - 48x + 65 15
Change to vertex form:
y = 7x 2 - 98x + 400 16
Change to vertex form:
y = 12x 2 - 60x + 312 17
Change to vertex form:
y = -5x 2 + 20x - 32 y = -5(x 2 - 4x + ___) - 32 - ___ y = -5(x 2 - 4x + 4 ) - 32 + 20 y = -5(x - 2) 2 - 12
Vertex at (2, -12)
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Change to vertex form:
y = -6x 2 + 72x - 53 y = -6(x 2 - 12x + ___) - 53 - ___ y = -6(x 2 - 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 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Complete the square to change to vertex form.
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Change to vertex form:
y
5
x
2 3
x
2
y
5
x
2 3
x
5 ___ 2 ___
y
5
x
2 3
x
5 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
7
x
2 9
x
25
y y
7
x
2 7
x
2 9 7
x
___ 9 7
x
81 196
y
7
x
9 14 2 619 28 25 ___ 25 81 28 9 14 , 619 28 22
Change to vertex form:
y
5
x
2 8
x
23
Change to vertex form:
y
1 2
x
2 2
x
3 24
x
2
x
2
Solve by completing the square.
4
x
5
x
2
2
4
x
5
x
2 2
9
9
x
2 4
x
__ 5 __
x
2 3
x
2
4
x
4
5
4
x
2
2
9
x
3 2
x
5 ,
1
25
2
x
2
Solve by completing the square.
5 12
x
x
3 2 2 2
x
x
2 2
x
2 2 12 6 6
x x x
5 __ 9 5 5 18 __ 2
x
3 2
x
3 2 23 2 23 23 2
x
3 2 23 2
x x
3 23 2
x
3 23 6 .
391 , 2 0 .
26 391
Example: Solve by completing the square
: x 2 + 6x – 8 = 0 x 2 + 6x - 8 = 0 x 2 + 6x = 8 x 2 + 6x + ___= 8 + ___ x 2 + 6x + 9 = 8 + 9 (x+3) 2 = 17
x
3 17
x
3 17 27
Solve by completing the square:
0
ax
2
bx
c ax
2
bx
c a
x
2
b a x
__
c
__
a
x
2
b a x
b
2 4
a
2
c
b
2 4
a a b
2
a
2
b
2 4
a
c
28
a
Solve by completing the square:
b
2
a
2
b
2 4
a
c b
2
a
2
b
2 4
a
2
c a x x
b
2
a
b
2
a
b
2 4
a
2
c a b
2 4
a
2
c a b
2
a
2
b
2 4
a
2
c a x
b
2
a
b
2 4
a
2 4
ac
4
a
2 29
Solve by completing the square:
x
b
2
a
b
2 4
a
2 4
ac
4
a
2
x
b
2
a
x
b
2
a
b
2 4
ac
4
a
2
b
2 4
ac
2
a x
b
b
2 4
ac
2
a
This is called the Quadratic Formula. You must memorize it!!!
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