1. Algebra 1
Section 12.4

2. 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.

3. 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.

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

5. 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

6. 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

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

8. 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

9. 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

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

11. 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 .

12. 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

13. 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

14. 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

15. Homework: pp