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Completing the Square

Completing the Square

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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 =

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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

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Change to vertex form:

y = 12x2 - 60x + 312

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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

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Change to vertex form:

xxy 85 2

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Change to vertex form:

322

1 2 xxy

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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!!!