Question 112309
We know that the area of a rectangle is given by {{{A=lw}}}.  But the problem tells us that {{{l=w+5}}}.  The length (l) is (=) 5 in (5) more than (+) its width (w).


Since we know that the area is 75 sq. in., we can now write {{{(w+5)*w=75}}}


Simplifying and solving:


{{{w^2+5w=75}}}
{{{w^2+5w-75=0}}}
This one doesn't factor, so use the quadratic formula
{{{w = (-5 +- sqrt( 5^2-4(-75)))/(2) }}}
{{{w = (-5 +- sqrt(325))/2}}}


{{{w = (-5 +- 5*sqrt(13))/2}}}


One of the possibilities, namely  {{{w = (-5 - 5*sqrt(13))/2}}}, yields a negative result which is an absurdity when you are trying to find the length of a rectangle's side, therefore, the only valid answer is {{{w = (-5 + 5*sqrt(13))/2}}}.  So now we have the width.


The length is just 5 inches longer, so {{{l = ((-5 + 5*sqrt(13))/2)+5}}}
or in simpler terms
{{{l=(5+5*sqrt(13))/2}}}


Leave your answers in terms of these simplest form expressions containing the radicals.  If the problem had asked you to determine the length and width to some number of decimal places accuracy, THEN you would use a calculator and round appropriately, but you have to presume that exact expressions for the lengths are required because the problem asks "what is the length and width" without an accuracy specification.