WS – Completing the Square
Algebra – accel.
11 April 2014
Recall that we can solve quadratic equations using the square root function:
𝑥 2 = 17
(𝑥 + 6)2 = 2
√𝑥 2 = ±√17
√(𝑥 + 6)2 = ±√2
𝑥 = ±√17
𝑥 + 6 = ±√2
𝑥 = −6 ± √2
Now we will use this technique in a process for solving quadratic equations that cannot be factored. The
process is called “completing the square” because it relies on creating a perfect square trinomial from the first
two terms of the quadratic trinomial to be factored. Here’s an example:
𝑥 2 − 4𝑥 − 7 = 0
solve this quadratic equation by completing the square
𝑥 2 − 4𝑥 = 7
move the constant term to the right side
𝑥 2 − 4𝑥 + 4 = 7 + 4
add 4 to both sides to complete the square on the left side
(𝑥 − 2)2 = 11
write the left-side trinomial in factored form
𝑥 − 2 = ±√11
square root both sides – don’t forget the ±
𝑥 = 2 ± √11
add 2 to both sides to solve for 𝑥
Practice these in class:
1. 𝑥 2 − 2𝑥 − 2 = 0
2. 𝑥 2 − 6𝑥 − 13 = 0
3. 𝑥 2 + 4𝑥 − 11 = 0
4. 𝑥 2 + 10𝑥 + 8 = 0
5. 𝑥 2 − 14𝑥 + 36 = 0
6. 𝑥 2 + 20𝑥 + 71 = 0
Homework: Solve these quadratic equations by completing the square:
7. 𝑥 2 − 10𝑥 + 23 = 0
8. 𝑥 2 − 2𝑥 − 9 = 0
9. 𝑥 2 + 6𝑥 + 4 = 0
10. 𝑥 2 + 2𝑥 − 5 = 0
11. 𝑥 2 − 8𝑥 + 5 = 0
12. 𝑥 2 + 12𝑥 + 17 = 0
13. 𝑥 2 − 10𝑥 − 6 = 0
14. 𝑥 2 − 20𝑥 − 81 = 0
15. 𝑥 2 + 16𝑥 + 34 = 0
16. 𝑥 2 + 24𝑥 + 111 = 0