Completing the Square

Solve

•

Remember how to simplify radicals?

Perfect Square Trinomials

Examples

x2 + 6x + 9

x2 - 10x + 25

x2 + 12x + 36

What pattern do you notice?

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

Steps to complete the square

• 1.) You will get an expression that looks like this:

AX²+ BX

• 2.) Our goal is to make a square such that we have

(a + b)² = a² +2ab + b²

• 3.) We take ½ of the X coefficient

(Divide the number in front of the X by 2)

• 4.) Then square that number

To Complete the Squarex2 + 6x

• Take half of the coefficient of ‘x’

• Square it and add it

3

9

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

Complete the square, and show what the perfect

square is:

xx 122 36122 xx 26x

yy 142 49142 yy 27y

yy 102 25102 yy 25y

xx 52 4

2552 xx

2

2

5

x

To solve by completing the square

• If a quadratic equation does not factor we can solve it by

two different methods

• 1.) Completing the Square (today’s lesson)

• 2.) Quadratic Formula (Next lesson)

Steps to solve by completing the square

1.) If the quadratic does not factor, move theconstant to the other side of the equation Ex: x²-4x -7 =0 x²-4x=7

2.) Work with the x²+ x side of the equation and complete the square by taking ½ of the coefficientof x and squaring Ex. x² -4x 4/2= 2²=4

3.) Add the number you got to complete the square toboth sides of the equationEx: x² -4x +4 = 7 +4

4.)Simplify your trinomial square Ex: (x-2)² =11

5.)Take the square root of both sides of the equationEx: x-2 =±√11

6.) Solve for xEx: x=2±√11

Solve by Completing the Square2 6 16 0x x

2 6 16x x +9 +9

2 6 9 25x x

2

3 25x

3 5x

3 5x 8x 2x

Solve by Completing the Square2 22 21 0x x 2 22 21x x

+121 +1212 22 121 100x x

2

11 100x

11 10x

11 10x 21x 1x

Solve by Completing the Square2 2 5 0x x

2 2 5x x +1 +1

2 2 1 6x x

2

1 6x

1 6x

1 6x

Solve by Completing the Square2 10 4 0x x

2 10 4x x +25 +25

2 10 25 29x x

2

5 29x

5 29x

5 29x

Solve by Completing the Square

01182 xx

1182 xx+16 +16

51682 xx

542x

54 x

54 x

Solve by Completing the Square

0462 xx

462 xx+9 +9

5962 xx

532x

53 x

53x

The coefficient of x2 must be “1”

0332 2 xx

02

3

2

32 xx

16

33

4

32

x

2

3

2

32 xx4

32

2

3

2 9 9

16 16

3 3

2 2x x

16

33

4

3x

16

33

4

3x

2 2 2 2

33

44

333x