Completing the Square

Grade 10Lesson 5-5

Completing the Square

This is an x. Show me x2.x x2

Show me x2 + 6x

Completing the Square

x2

x2

x x xx x x

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!

Let’s Try Another One!

Show me x2 + 2x

x2 x x

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!

Last One with Manipulative

Show me x2 + 8x

x2 x xx x x x x x

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

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?

What is completing the square used for?

Completing the square is used for all those not factorable problems!!

It is used to solve these equations for the variable.

Rule for Completing the Square

bxx 2

22

2

b

bxx

2

2

bx

This is now a PTS!So, it factors into

this!

Example: Find the value of c that makes this a PTS, then write the expression as the square of a binomial. x2-3x+c

b=-3

22

2

3

2

b

c

4

9

2

2

3

x

4

932 xx

Example: Solve by completing the square. x2+6x-8=0

x2+6x-8=0x2+6x=8x2+6x+___=8+___

x2+6x+9=8+9(x+3)2=17

932

6

22

22

b

17 3 x

173x

Don’t forgetDon’t forget: Whatever you add to one side of an equation, you MUST add to the other side!

More Examples!5x2-10x+30=0x2-2x+6=0

x2-2x=-6x2-2x+__=-6+__

x2-2x+1=-6+1(x-1)2=-5

3x2-12x+18=0x2-4x+6=0x2-4x=-6x2-4x+__=-6+__

x2-4x+4=-6+4(x-2)2=-2

112

2

22

22

b

51 x

51 ix

422

4

22

22

b

22 x

22 ix

Last Example! Write the quadratic function y=x2+6x+16 in vertex form. What is the vertex of the function’s graph?

y=x2+6x+16y-16=x2+6xy-16+__=x2+6x+__

y-16+9=x2+6x+9y-7=(x+3)2

y=(x+3)2+7

If the equation, in vertex form, is y=(x+3)2+7, then the vertex must be (-3,7). 93

2

6

22

22

b

Solving Quadratic Equations by

Completing the Square

Perfect Square Trinomials

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

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

Perfect Square Trinomials

Create perfect square trinomials.

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

100

4

25/4

Solving Quadratic Equations by Completing the Square

Solve 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

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.

2 8 =20 + x x 21

( ) 4 then square it, 4 162

8

2 8 2016 16x x

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

Solving Quadratic Equations by Completing the Square

Step 4: Take the square root of each side

2( 4) 36x

( 4) 6x

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

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

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

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

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

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