Question 130770
{{{20^2 = x^2 + (x + 4)^2}}} looks mighty good to me.  Let's see where this takes us:


I'll start by turning things around:
{{{x^2 + (x + 4)^2=20^2}}}


Expand and collect terms:
{{{x^2+(x^2+8x+16)=400}}}


{{{2x^2+8x+16=400}}}


Put in standard form ({{{ax^2+bx+c=0}}}
{{{2x^2+8x+16-400=0}}}


{{{2x^2+8x-384=0}}}


Divide by the lead coefficient to get things in lowest terms:
{{{x^2+4x-192=0}}}


Finally, factor the trinomial (or complete the square, or use the quadratic formula, according to your preference):
{{{(x-12)(x+16)=0}}}, because {{{-12*16=-192}}} and {{{-12+16=4}}}


This says that {{{x=12}}} or {{{x=-16}}}, but we can exclude the negative root for two reasons.  First, because we are looking for a measure of length, a negative number is absurd, and secondly, since in setting up the original equation you squared the variable, you introduced an extraneous root -- one that satisfies the derived equation, but does not satisfy the original problem.


Therefore, the correct answer is that the short leg is 12 feet and the long leg is 16 feet.


Check the answer:

{{{12^2=144}}}, {{{16^2=256}}}, {{{144+256=400}}}, and {{{sqrt(400)=20}}}.  Answer checks.