Completing the Square

Mathematics · Algebra · ages 15-16

Name ______________________ Date ____________

In the form (x + p) squared + q for x squared + 8x + 3, what is p?
Answer: ______________
Completing the square on x squared + 4x + 7 adds and subtracts 4.
Circle one: True False
Write x squared + 4x + 9 as (x + p) squared + q. What is q?
- 13
- 5
- -5
- 17
Which is x² - 8x + 3 in completed square form?
- (x - 4)² + 3
- (x + 4)² - 13
- (x - 8)² - 61
- (x - 4)² - 13
The form (x + 4) squared - 3 was completed from x squared + bx + c. What is b?
- -8
- 4
- 8
- -4
Solve x squared + 4x - 5 = 0 by completing the square.
- x = 1 or x = -5
- x = 1 or x = 5
- x = -1 or x = -5
- x = -1 or x = 5
Solve x² - 10x + 25 = 0.
Answer: ______________
Write x squared + 12x + 5 as (x + p) squared + q.
- (x + 6) squared - 31
- (x + 12) squared - 139
- (x + 6) squared + 41
- (x + 3) squared - 31
Ben writes 2x squared + 8x + 3 as 2(x + 2) squared - 5, then solves 2x squared + 8x + 3 = 0 by dividing by 2 to get (x + 2) squared - 5 = 0. What is wrong with that step?
- The completed square should be 2(x + 4) squared - 13
- Dividing by 2 must divide the constant too, so it is (x + 2) squared - 5/2 = 0
- He should have divided by 4
- Nothing went wrong
Write 3x squared + 6x + 1 as a(x + p) squared + q.
- 3(x + 1) squared - 2
- 3(x + 2) squared - 5
- 3(x + 1) squared - 8
- 3(x + 2) squared - 2

Answer key

For grown-ups. Fold this page away before handing over the rest.

4 · Half of 8 is 4, so the square is (x + 4) squared, and p is 4.
True · Half of 4 is 2, and 2 squared is 4, so completing the square on x squared + 4x + 7 does add and subtract 4.
5 · Add and subtract 4: x squared + 4x + 4 - 4 + 9 is (x + 2) squared + 5, so q is 5.
(x - 4)² - 13 · Half of -8 is -4, (-4)² is 16, and 3 - 16 = -13.
8 · Expanding (x + 4) squared gives x squared + 8x + 16, so the x coefficient b is 8.
x = 1 or x = -5 · Complete the square: (x + 2) squared - 9 = 0, so (x + 2) squared = 9, giving x + 2 = plus or minus 3, so x = 1 or x = -5.
5 · The left side is (x - 5)², so x = 5 (a repeated root).
(x + 6) squared - 31 · Add and subtract 36: x squared + 12x + 36 - 36 + 5 is (x + 6) squared - 31.
Dividing by 2 must divide the constant too, so it is (x + 2) squared - 5/2 = 0 · His completed square is right. Dividing the whole equation by 2 gives (x + 2) squared - 5/2 = 0, not (x + 2) squared - 5 = 0.
3(x + 1) squared - 2 · Take out 3: 3(x squared + 2x) + 1. Inside, add and subtract 1: 3((x + 1) squared - 1) + 1, which is 3(x + 1) squared - 3 + 1, that is 3(x + 1) squared - 2.