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Algebra - xy=8 and x-y=6

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by moshum » Wed Jan 14, 2009 7:16 am
if x-y=6 then (x-y)(x-y)=36

(x-y)(x-y)= x^2 - 2xy + y^2 = 36

since we know that xy = 8 it follows:

x^2 - 2(8) + y^2 = 36

so x^2 + y^2 = 52

Answer is (E)
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by Brent@GMATPrepNow » Wed Jan 14, 2009 7:19 am
Nice work, Moshum. The correct answer is E.

I've pasted my solution as well:
We can quickly see that substitution does not work here.
From x-y=6, we conclude that x=y+6 and we substitute this into xy=8 to get (y+6)y = 8, which simplifies to be y^2 + 6y – 8 = 0. The quadratic cannot be factored. So, now what?
We can use the fact that (x-y)^2 = x^2 – 2xy + y^2
Notice that the result (x^2 – 2xy + y^2) is very close to the x^2 + y^2 we are looking for.
Okay, so here we go:
(x-y)^2 = x^2 – 2xy + y^2
6^2 = x^2 – 2xy + y^2 (replace x-y with 6)
6^2 = x^2 – 2(8) + y^2 (replace xy with 8)
36 = x^2 – 16 + y^2 (simplify)
52 = x^2 + y^2 (solve for x^2 + y^2)
The correct answer is E
Brent Hanneson - Creator of GMATPrepNow.com
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