1. MODELING COMPLETING THE SQUARE
Use algebra tiles to complete a perfect square trinomial.
Model the expression x 2 + 6x. x2 x x x x x x
Arrange the x-tiles to form part of a square. x 1 1 1 x 1 1 1
To complete the square, add nine 1-tiles. x 1 1 1
x2 + 6x + 9 = (x + 3)2
You have completed the square.

2. ( ) b x2 + bx + = x + 2 SOLVING BY COMPLETING THE SQUARE
To complete the square of the expression x2 + bx, add the square of half the coefficient of x.
x2 + bx = x + ) ( b 2

3. ( ) ( ) –8 2 –8 x2 – 8x + = x2 – 8x + 16 = (x – 4)2 2
Completing the Square
What term should you add to x2 – 8x so that the result is a perfect square?
SOLUTION
The coefficient of x is –8, so you should add , or 16, to the expression. ) ( –8 2
x2 – 8x + ) ( –8 2
= x2 – 8x + 16 = (x – 4)2

4. ( ) ( ) 2x2 – x – 2 = 0 2x2 – x = 2 x2 – x = 1 x2 – x + = 1 + 1 2 1 2
Completing the Square
Factor 2x2 – x – 2 = 0
SOLUTION
2x2 – x – 2 = 0
Write original equation.
2x2 – x = 2
Add 2 to each side.
x2 – x = 1 1 2
Divide each side by 2.
Add = , or ( 1 2 – • ) 16 4
to each side. 1 2
x2 – x = 1 + ) ( 4 – 16

5. Completing the Square 1 2
x2 – x = 1 + ) ( 4 – 16
Add = , or •
to each side. ) 1 4 2 – ( x = 17 16
Write left side as a fraction. 17 4 1
x – =
Find the square root of each side.
x = 1 4 17
Add to each side. 1 4 1 4
The solutions are +  and  – 0.78. – 17

6. 17 The solutions are +  1.28 and  – 0.78. 1 – 4
Completing the Square 1 4
The solutions are +  and  – 0.78. – 17
CHECK
You can check the solutions on a graphing calculator.

7. ( ) ( ) ( ) ax2 + bx = – c – c x2 + + = – c 2 x2 + + = + 2 – c x + = +
CHOOSING A SOLUTION METHOD
Investigating the Quadratic Formula
Perform the following steps on the general quadratic equation
ax2 + bx + c = 0 where a  0.
ax2 + bx = – c
Subtract c from each side.
x = bx a
– c
Divide each side by a. bx a
x = b 2a ) (
– c 2
Add the square of half the coefficient of x to each side. b 2a ) ( 2
x =
– c a b2
4a2
Write the left side as a perfect square. b 2a ) ( 2
x =
– 4ac + b 2
4a2
Use a common denominator to express the right side as a single fraction.

8. ( ) ± - 4 ac ± - 4 ac ± b - 4 ac 2 – 4ac + b 2 x + = x + = x = – –b
CHOOSING A SOLUTION METHOD
Investigating the Quadratic Formula
Use a common denominator to express the right side as a single fraction. b 2a ) ( 2
x =
– 4ac + b 2
4a 2 b 2a 2 - 4 ac
x =
Find the square root of each side. Include ± on the right side.
x = – b 2a 2 - 4 ac
Solve for x by subtracting the same term from each side.
x = 2a –b b 2 - 4 ac
Use a common denominator to express the right side as a single fraction.