PrepSwift Finding the Minimum Value: Q2 via Completing the Square

GRE Quant Problem Solving
1

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Each time I solve this, I am ending up with -(x + .5)^2 + .75 when it should be +1.25

Here is what I’ve been doing–

  1. Original equation: 1 - x - x^2
  2. I am re-writting it to y= -x^2 - x + 1
  3. Factor out the negative to get y = - (x^2 + x) + 1
  4. Completing the square gives me y + .25 = - (x^2 + x + .25) + 1
  5. Solving: y = - (x + .5)^2 + .75

I can tell that I am making a mistake somewhere with positive/negative, but I can’t pinpoint where or why I am making that mistake.

EDIT: I understand that solving this problem isnt necessary, but I wanted to do so anyway to get the practice.

2

Try again.

3

Sadly I’ve been trying again for over an hour. IIRC, we should add and subtract the .25 from both sides, which would still just leave me with 1.

4

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Following this guidance, when I complete the square I have

y + .25 = -(x^2 + x + .25) + 1

Is there an interaction with the factored -1 that should cause me to write it as y - .25 = -(x^2 + x + .25) + 1?

5

On the LHS you’re actually adding, while on the RHS you’re subtracting. Watch out for the minus on the RHS.

You seem to have a vague understanding of the problem, but this is not correct.

6

I wanted to solve it just for the repetition, acknowledge its not needed to answer this question. I see that I need to be careful with how the factored - interacts with (x^2 +x) (does not change it to negative on right) but does change it on the left.

7

Did you now see the mistake you made in the original problem?

8

Yes, I believe that since I factored out a - to get (x^2 + x), when I complete the square and “add” the .25 to get -(x^2 +x + .25) I am actually adding a negative (since the factored negative applies to the constant, but not the square), which means I would actually subtract it from the other side in order to make it equivalent.

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