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.

( ) 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

( ) ( ) –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

( ) ( ) 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

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 +  1.28 and  – 0.78. – 17

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

( ) ( ) ( ) 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. x2 + + = bx a – c Divide each side by a. bx a x2 + + = + 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.

( ) ± - 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.