1

1

Completing the Square

This is an x. Show me x2.

x x2

Show me x2 + 6x

2

2

Completing the Square

x2

x2

x x xx x x

3

3

Completing the Square

The picture is now x2 + 6x + 9,

How many units are required to complete the square?

which factors (x+3)(x+3) = (x+3)2

9!

4

4

Let’s Try Another One!

Show me x2 + 2x

x2 x x

5

5

Completing the Square

x2

How many units are required to complete the square?

x2

The picture is now x2 + 2x + 1

which factors (x+1)(x+1) = (x+1)2

1!

6

6

Last One with Manipulatives

Show me x2 + 8x

x2 x xx x x x x x

7

7

Completing the Square

Again, how many units are required to complete the square?

16

So, the picture is now x2 + 8x + 16

which factors (x+4)(x+4) = (x+4)2

8

8

Hard One!

Complete the square for x2 + 18x + ___

How many units are needed?

There are not enough pieces to do this problem.

Can we do it using paper and pencil?

9

9

Perfect Square Trinomials

Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36

10

10

Creating a Perfect Square Trinomial

In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____

Find the constant term by squaring half the coefficient of the linear term.

(14/2)2

X2 + 14x + 49

11

11

Perfect Square Trinomials

Create perfect square trinomials.

x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___

100

4

25/4

12

12

Solving Quadratic Equations by Completing

the SquareSolve the following

equation by completing the square:

Step 1: Move quadratic term, and linear term to left side of the equation

2 8 20 0x x

2 8 20x x

13

13

Solving Quadratic Equations by Completing

the SquareStep 2: Find the term

that completes the square on the left side of the equation. Add that term to both sides.

2 8 =20 + x x 21

( ) 4 then square it, 4 162

8

2 8 2016 16x x

14

14

Solving Quadratic Equations by Completing the Square

Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

2 8 2016 16x x

2

( 4)( 4) 36

( 4) 36

x x

x

15

15

Solving Quadratic Equations by Completing the Square

Step 4: Take the square root of each side

2( 4) 36x

( 4) 6x

16

16

Solving Quadratic Equations by Completing the Square

Step 5: Set up the two possibilities and solve

4 6

4 6 an

d 4 6

10 and 2 x=

x

x x

x

17

17

Completing the Square-Example #2

Solve the following equation by completing the square:

Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation.

22 7 12 0x x

22 7 12x x

18

18

Solving Quadratic Equations by Completing the Square

Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.

The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first.

2

2

2

2 7

2

2 2 2

7 12

7

2

=-12 +

6

x x

x x

xx

21 7 7 49

( ) then square it, 2 62 4 4 1

7

2 49 49

16 1

76

2 6x x

19

19

Solving Quadratic Equations by Completing the Square

Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

2

2

2

76

2

7 96 49

4 16 16

7 47

4

49 49

16 1

16

6x x

x

x

20

20

Solving Quadratic Equations by Completing the Square

Step 4: Take the square root of each side

27 47( )

4 16x

7 47( )

4 4

7 47

4 4

7 47

4

x

ix

ix

21

21

Solving Quadratic Equations by Completing the Square

2

2

2

2

2

1. 2 63 0

2. 8 84 0

3. 5 24 0

4. 7 13 0

5. 3 5 6 0

x x

x x

x x

x x

x x

Try the following examples. Do your work on your paper and then check your answers.

1. 9,7

2.(6, 14)

3. 3,8

7 34.

2

5 475.

6

i

i

22

22

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