Question 92180
Well, your initial idea is right on!
{{{(x-3)^2 = A-81}}} Where A = original area and x = the original length of the side of the square.
The next step is to recall that {{{A = x^2}}}, that is; the original area (A) is equal to the original sides squared {{{(x^2)}}}, so substitute the A with {{{x^2}}} in the first equation then solve for x.
{{{(x-3)^2 = x^2-81}}} Expand the left side.
{{{x^2-6x+9 = x^2-81}}} Subtract {{{x^2}}} from both sides.
{{{-6x+9 = -81}}} Subtract 9 from both sides.
{{{-6x = -90}}} Finally, divide both sides by -6.
{{{x = 15}}}cm.

Check:
{{{(15-3)2 = 15^2-81}}}
{{{12^2 = 225-81}}}
{{{144 = 144}}}