Completing the Square

1

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

3

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

11

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.

12

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)

13

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)

14

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)

18

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)

19

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.

20

   

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)

21

   

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!!!

30