Algebra 1 Section 12.4

Completing the Square Given ax2 + bx + c = 0, where a ≠ 1, you will need to first divide both sides of the equation by a, the leading coefficient.

Begin by dividing both sides by 4, the leading coefficient. Example 1 Solve 4x2 + 16x – 8 = 0. Begin by dividing both sides by 4, the leading coefficient. x2 + 4x – 2 = 0 x2 + 4x = 2 Add 4 to both sides.

Example 1 x2 + 4x + 4 = 2 + 4 x2 + 4x + 4 = 6 (x + 2)2 = 6 x + 2 = ± 6

Begin by dividing both sides by 3, the leading coefficient. Example 2 Solve 3x2 – 2x – 1 = 0. Begin by dividing both sides by 3, the leading coefficient. x2 – x – = 0 2 3 1 x2 – x = 2 3 1

The number that completes the square is 1/9. Example 2 x2 – x = 2 3 1 The number that completes the square is 1/9. x2 – x + = + 2 3 1 9 (x – )2 = 1 3 4 9

Example 2 (x – )2 = 1 3 4 9 x – = ± 1 3 2 x = ± 1 3 2 x = 1, - 1 3

Divide both sides by 2, the leading coefficient. Example 3 Solve 2x2 = 10x + 3. 2x2 – 10x = 3 Divide both sides by 2, the leading coefficient. x2 – 5x = 3 2

Complete the square by adding (5/2)2 = 25/4 to each side. Example 3 x2 – 5x = 3 2 Complete the square by adding (5/2)2 = 25/4 to each side. x2 – 5x + = + 25 4 3 2 (x – )2 = 5 2 31 4

Example 3 (x – )2 = 5 2 31 4 x – = ± 31 2 5 x = ± 31 2 5 x = 5 ± 31 2

Projectile Motion The motion of a projectile with an initial vertical velocity vi (in ft/sec) and initial height hi is modeled by the equation h(t ) = -16t 2 + vi t + hi .

Example 4 h(t ) = -16t 2 + vi t + hi 15 = -16t2 + 88t + 3 Divide both sides by 16. t 2 – x = - 11 2 3 4

Half of 11/2 is 11/4, and when you square it, you get 121/16. Example 4 t 2 – x = - 11 2 3 4 Half of 11/2 is 11/4, and when you square it, you get 121/16. t 2 – t + = - + 11 2 121 16 3 4 (t – )2 = 11 4 109 16

Example 4 (t – )2 = t – = ± t = t ≈ 5.36, 0.14 sec 11 4 109 16 109 4 11 ± 109 4 t ≈ 5.36, 0.14 sec

Homework: pp. 501-502