Question

How do you find the value of c that makes `x^2-24x+c` into a perfect square?

Answer 1

Divide the coefficient of `x` (not `x^2`) by 2, and then square that.
`c=144`.

Explanation:

Divide the coefficient of `x` (not `x^2`) by 2, and then square that.

So for `x^2-24x`, dividing `-24` by `2` gets us `-12`.
Squaring `-12` gets us `144`,

So `c=144`.

---

In general:

Consider the factoring identity

`\qquad(x-a)^2 = x^2 - 2ax+ a^2`

If we only have the middle term, `- 2ax`, then dividing the middle term by `2x` gets us

`\qquad\frac{-2ax}{2x} = -a`

(If we only take the coefficient on `x`, we do not have to divide by `x`.)

Then if we square that, we get

`\qquad(-a)^2 = a^2`

So as you can see, taking the coefficient on `x`, dividing it by `2`, and then squaring it will get us the magic constant that lets us complete the square.