Question 100153
The statement that the "length is 1 cm longer than the width" means that we can define the length (l) = w + 1.  Width (w) is simply = to w. So, the rectangle has sides of w, w+1, w, and w+1.  The diagonal is known to be 4 cm (given).

Now we simply apply the Pythagorean formula. Let's call the diagonal d.  That means 

{{{d^2 = w^2 + (w+1)^2}}}

{{{(w+1)^2 = w^2 + 2w + 1}}}

Therefore, {{{d^2 = w^2 + w^2 + 2w + 1 = 2w^2 + 2w + 1}}}

Of course, {{{d^2}}} is the square of the diagonal, and we were told the diagonal was 4 cm, so {{{d^2 = 16}}}.

Substituting what we know:

{{{16 = 2w^2 + 2w + 1}}}

Subtracting 16 from both sides:

{{{0 = 2w^2 + 2w - 15}}}

This does not factor easily, so you can use the quadratic equation to find w.

Quadratic formula: {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}

Substituing, we have {{{w = (-2 +- sqrt( 2^2-4*2*(-15) ))/(2*2) }}}