Solving Quadratic Equations by

Completing the Square

Perfect Square Trinomials

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

Rule for Completing the Square

bxx 2

22

2

b

bxx

2

2

bx

This is now a PST!So, it factors into

this!

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

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=0

x2-2x+6=0x2-2x=-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

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