Question

How do you complete the square for `x^2-4x`?

Answer 1

`x^2-4x=(x-2)^2-4`

Explanation:

To complete the square, we need to find some binomial, `a+b`, such that `(a+b)^2` only differs by a constant to the expression we want.

Since `(a+b)^2=a^2+2ab+b^2`, `a` must be equal to `x`. Now we only need to find `b`:
`(x+b)^2=x^2+2bx+b^2`.

Since we want the middle term to be `-4x`, we know that `2b=-4`. Dividing both sides by two, we get:
`b=-4/2=-2`.

Now we have a square that will get us the first two terms, but we also will have an extra bit at the end, which isn't right:
`(x-2)^2=x^2-4x+4`

Since we want no constant at all, we just have to subtract by `4`:
`(x-2)^2-4`

We can expand it just to check:
`(x-2)^2-4=x^2-4xcancel(+4-4)=x^2-4x`