Question 1096124
<br>If the length of the decorative area is 3 more than the width, and the border is uniform width, then the length of the whole rug will still be 3 more than the width.  So the problem is easier to solve if you just solve the equation<br>
{{{x(x+3) = 108}}}
{{{x^2+3x = 108}}}
{{{x^2+3x-108 = 0}}}
{{{(x+12)(x-9) = 0}}}<br>
x = -12 or x = 9; obviously the positive answer is the one that makes sense.<br>
Since the problem asks for the area of the rug including the border, we know the width is 9 and the length is x+3 = 12.<br>
Note that solving the problem algebraically involves solving a quadratic equation; to do that by factoring, we need to find two numbers whose difference is 3 and whose product is 108.<br>
But that is exactly what the problem asks us to do, WITHOUT doing any algebra: find a length and a width whose product is 108, with the length 3 more than the width.<br>
So using formal algebra to solve this problem is a waste of time -- except as practice in solving problems using algebra.