Question 135595
length l times width w = area A

{{{l*w = A}}}

{{{l*w = 52cm^2}}}

Also we know that the length is 1cm more than 3 times the width.

{{{l = 3*w + 1}}}

Plugging that back into the Area equation, we get:

{{{(3*w + 1) *(w) = 52cm^2}}}
Simplify
{{{3*w^2 + w = 52}}}
{{{3*w^2 +w - 52 = 0}}}
{{{(3*w+13)*(w-4)=0}}}
Two Real solutions!!
Either {{{3*w +13=0}}} or {{{w - 4 = 0}}}
Either {{{3*w = -13}}} or {{{w = 4}}}
Either {{{w = -13/3}}} or {{{w = 4}}}

Because having a negative answer for a length doesn't make sense, we can disregard the -13/3 answer for w.  So, width is 4.  Length is 3 times width plus one, or 13.  

Perimeter is {{{2*l + 2*w}}} - will leave that to the student to answer.