x2 + 10x must have come from squaring (x + 5), so 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 whatever we're squaring, it will be riddled with question marks and look like
(?x + ?).
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. The thing we're squaring looks like (2x + ?).
To find -12x after squaring, the product 2x(?) must be -6x. Since we're squaring, the product of the outside terms will be the same as the product of the inside terms, so each product must be half of -12x. The missing number must be -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.