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December 19, 2014

December 19, 2014

Posted by **luke** on Thursday, May 3, 2007 at 9:21pm.

if x^3-y^3=m and x-y=N, express xy in terms of M and N

This is one of those pure manipulation problems.

you have to know that x^3-y^3 = (x-y)(x^2+xy+y^2)

so n(x^2+xy+y^2)=m

(x^2+xy+y^2) = m/n (#1)

notice we have xy, which is what we want, inside the bracket, but there is that x^2+y^2 hanging around

(from countless years of experience I know)

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

so let's put that back in #1

((x-y)^2 + 2xy + xy) = m/n

n^2 + 3xy = m/n

3xy = m/n - n^2

3xy = (m-n^3)/n

and finally

xy = (m-n^3)/(3n)

I don't know what grade level you are in but that is pretty nasty.

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