Completing the Square

Solving Quadratics By

Completing the SquarePart 2Must be a

perfectSquare

2( 5) 64x 5 8x

When you take the square root, You MUST consider the Positive and Negative answers.

5 8x 5 8x 5 5

13x 5 5

3x

PerfectSquare

On One side

Take Square Root

ofBOTH SIDES

2 ( 5) 64x

PerfectSquare

On One side

Take Square Root

ofBOTH SIDES

But what happens if you DON’T have a perfect square on one side…….

You make it a Perfect Square

Use the relations on next slide…

2( 6)x ( 2 ) To expand a perfect square binomial:

2 12 36x x 6x 26

We can use these relations to find the missing term….To make it a perfect square trinomial that can be factored into a perfect square binomial.

2 _ _12 _x x 12 2 6 626 36

36

2x

Take ½ middle term

Then square it

The resulting trinomial is called a perfect square trinomial,

which can be factored into a perfect square binomial.

2 _ _18 _ _x x

18 2 92(9) 81

81 2( 9)x

1. 2 12 0x x

1. Make one side a perfect square

2. Add a blank to both sides

3. Divide “b” by 2

4. Square that answer.

5. Add it to both sides

6. Factor 1st side

7. Square root both sides

8. Solve for x

2 0x x ___ ___ 12 2 6

2(6) 36

36 362( 6)x 362( 6) 36x 6 6x

6 6x 6 6x 6 6

12x 6 6

0x

12

Factor this Perfect square trinomial2 12 36x x

What is the

Square root

of x2

2( )x

Bring dow

n sign

6W

hat is the S

quare root of 36

2( 6)x

2. 2 8 0x x

1. Move constant to other side.

2. Add a blank to both sides

3. Divide “b” by 2

4. Square that answer.

5. Add it to both sides

6. Factor 1st side

7. Square root both sides

8. Solve for x

2 8x x ___ ___6 2 3

2(3) 9

9 92( 3)x 12( 3) 1x

3 1x 3 1x 3 1x 3 3

4x 3 3

2x

6

6

Factor this Perfect square trinomial2 6 9x x

What is the

Square root

of x2

2( )x

Bring dow

n sign

3W

hat is the S

quare root of 9

2( 3)x

3. 2 8 84 0x x

1. Move constant to other side.

2. Add a blank to both sides

3. Divide “b” by 2

4. Square that answer.

5. Add it to both sides

6. Factor 1st side

7. Square root both sides

8. Solve for x

2 84x x ___ ___8 2 4

2(4) 1616 16

2( 4)x 1002( 4) 100x

4 10x 4 10x 4 10x 4 4

14x 4 4

6x

8

Factor this Perfect square trinomial2 8 16x x

What is the

Square root

of x2

2( )x

Bring dow

n sign

4W

hat is the S

quare root of 9

2( 4)x

4. 2 2 15 0x x

1. Move constant to other side.

2. Add a blank to both sides

3. Divide “b” by 2

4. Square that answer.

5. Add it to both sides

6. Factor 1st side

7. Square root both sides

8. Solve for x

2 15x x ___ ___2 2 1

2(1) 11 1

2( 1)x 162( 1) 16x

1 4x 1 4x 1 4x 1 1

3x 1 1

5x

2

Factor this Perfect square trinomial2 2 1x x

What is the

Square root

of x2

2( )x

Bring dow

n sign

1W

hat is the S

quare root of 9

2( 1)x

Steps to solve Quadratics by completing the square:

Move the constant to side by itself. Make the side (w/variables) a perfect square by

adding a certain number to both sides. To calculate this number

– Divide “b” (middle term) by 2– Then square that answer

Take the square root of both sides of eq Then solve for x

In a perfect square, there is a relationship between the coefficient of the middle term and the constant term.

2( 7)x

7 1 (14)2

27 49

2 14 49x x

7