Question 916118
Dimensions x and y;
p=66
xy=270
Use A=xy


The question happens often as an academic exercise.
A, p known.
x, y unknown.


2x+2y=p and xy=A.
{{{2y=p-2x}}}
{{{y=(p-2x)/2}}}
-
{{{x(p-2x)/2=A}}}
{{{x(p-2x)=2A}}}
{{{xp-2x^2=2A}}}
{{{-2x^2+xp-2A=0}}}
{{{2x^2-xp+2A=0}}}


You can continue symbolically like this and use formula for solution of a quadratic equation, or you can substitute the given values for p and A now, and reach your values for x and y.


{{{2x^2-66x+2*270=0}}}
{{{x^2-33x+270=0}}}


{{{x=(33+- sqrt(33^2-4*270))/2}}}
{{{x=(33+- sqrt(9))/2}}}
{{{x=(33+- 3)/2}}}
  

The dimensions are 15 and 18.