Math PicnicFree printable practice

Completing the square worksheets

Completing the square rewrites a quadratic as a square plus a number. These free printable worksheets ask for both, and the answer key shows the step everybody skips: the square you build brings a number with it that has to be given back.

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Completing the square
Completing the square worksheet — Monic, whole answers, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

“Halve it and square it” is the half everyone remembers and it is only half the method. (x − 3)² is x² − 6x + 9, so it has quietly added a 9 that the original expression did not have, and the last step is subtracting it again. A student who writes (x − 3)² − 31 instead of (x − 3)² − 40 has done the memorable half and nothing else — which is why every key here expands the bracket before it touches the constant.

Why this sheet works

Every middle coefficient is even, and that is not a limitation being hidden. An odd one completes to a fraction — x² + 5x + 1 is (x + 5/2)² − 21/4 — which is a real question and not one a whole-number answer box can take. Every exercise set at this stage is even for the same reason.

How this one works

One question from this sheet, worked through a step at a time.

x² + 2x − 2

2 ÷ 2 = 1
(x + 1)² = x² + 2x + 1
x² + 2x − 2 = (x + 1)² − 3
  1. Halve the middle coefficient. Half of 2 is 1, and that number goes straight into the bracket — sign and all.
  2. But (x + 1)² is not the expression yet. Multiply it out and it brings a 1 along with it, and the original has −2 there instead.
  3. So give the 1 back: −2 − 1 = −3. That is the number that goes after the bracket.

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