Completing the Square

for Conic Sections

The Aim of Completing the Square

… is to write a quadratic function as a perfect square. Here are some examples of perfect squares!

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

Try to factor these (they’re easy).

Perfect Square Trinomials

x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36Can you see a numerical connection

between …6 and 9 using 3

-10 and 25 using -5

12 and 36 using 6

=(x+3)2

=(x-5)2

=(x+6)2

For a perfect square, the following relationships will always be true …

x2 + 6x + 9

x2 - 10x + 25

The Perfect Square Connection

Half of these values squared … are these values

The Perfect Square Connection

In the following perfect square trinomial, the constant term is missing. Can you predict what it might be?

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 the constant term (i.e. the number) to right 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

For chapter 10 material, we can stop here. But solving is a simple process from here …

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 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

Use calculator to do

this!Use calculator to do this!Use calculator to do this!Use calculator to do this!Use calculator to do this!