Call the length of the rectangle x. Then w = 2x - 7. The area is then

A = x(2x - 7) = 2x^2 - 7x. The area is known to be 23 cm^2 (Note the unit change, you had it wrong in the problem statement.) So:

23 = 2x^2 - 7x

2x^2 - 7x - 23 = 0

By the quadratic formula:

x = [-(-7) (+/-) sqrt{(-7)^2 - 4 * 2 * (-23)}]/(2 * 2)

x = [7 (+/-) \sqrt{233}]/4 (after a bit of work.)

Note: x must be positive, so we can only have the "+" solution here.

Now, we want the diagonal. This is going to be

sqrt{x^2 + (2x - 7)^2} = 5x^2 - 28x + 49 (by the Pythagorean theorem.)

sqrt{5 [ (7 + sqrt{233})/4 ]^2 - 28 [ (7 + sqrt{233})/4 ] + 49}

sqrt{5 [ 141/6 + 7*sqrt{233}/8 ] - 28 [ (7 + sqrt{233})/4 ] + 49}

sqrt{ (705 - 21*sqrt{233})/4

-Dan