Completing the Square Examples

If x² + 10x + c is a square, what is c?

The coefficient on the x term is 10, so divide 10 by 2 and square it:

10 ÷ 2 = 5

5² = 25

Yeah, that's the stuff: c = 25 is the missing constant term.

Assuming the expression is a square, find c in the expression 4x² – 12x + c. What did we need to square? Don't worry, c. We're coming for you.

What can we square to find 4x² – 12x + c? We know that whatever we're squaring is gonna look like (px + q). We'll need to work backwards, so turn around in your chair and bend your arms behind you. This will be tricky.

We know that the coefficient on the x term must be 2 for us to find 4x² after squaring. That's because we memorized our list of squares, thank you very much. So p = 2, and the thing we're squaring looks like (2x + q).

Since our middle term is -12x after squaring, we know that 2x(q) + 2x(q) must be -12x.

2x(q) + 2x(q) = -12x

4xq = -12x

The missing number must be q = -3. That's because we memorized our basic multiplication table, thank you very much. (What do you mean, we sound kind of defensive?) Time to check:

(2x – 3)² = 4x² – 12x + 9

The coefficients on the x² and x terms work out, and now we know that c = 9. We had to square the term (2x – 3) to find it, but find it we did.