Nia H. answered • 08/03/24
Mathematician, Programmer, and Composer for Tutoring
If we have an expression like (x + a)2, we know that this expands via FOIL as (x + b)2 = (x + b)(x + b) = x2 + bx + xb + b2 = x2 + 2bx + b2. What we want to do to complete the square is to get -2x2 + 8x - 8 = 0 into a form similar to x2 + 2bx + b2 so we can go backwards and turn it back into an (x + b)2 form (thus "completing the square" as we made it a single parentheses term being squared).
Since the right side is equal to 0, we can divide by -2 without changing the right side as 0 / -2 = 0, and that's going to make our life easier as notice how x2 + 2bx + b2 has no coefficients on x2 (it's not -2x2: it's just x2). Dividing by -2, we have the equivalent equation x2 - 4x + 4 = 0. Since we have a -4x now instead of +4x, let's recall that we can also complete the square by getting the equation in the form of (x - b)2 = x2 - bx - xb + (-b) * (-b) = x2 - 2bx + b2 (notice the x - b instead of x + b). Matching up terms, we see that -4 = -2b, so dividing by -2 we have b = 2.
We next need to match up b2 to what we have in the original equation. In the original equation---the one we got after dividing by -2---we have x2 - 4x + 4 = 0. The good news for us is in this problem, b2 = 22 = 4, so we already have the + 4 and the + b2 matching! In some problems, you'll have to do an extra step here where you add whatever you're missing, but we don't need to discuss this extra step right now as this problem worked out nicely.
Showing the problem in total where we complete the square, we have:
-2x2 + 8x - 8 = 0
x2 - 4x + 4 = 0 (dividing by -2 as we showed earlier)
(x - 2)2 = 0 (because we know (x - 2)2 = x2 - 4x + 4 from our earlier work)
The only solution is x = 2 as 2 - 2 = 0.
