The Width of rectangle

Question

The width of a rectangle is 1 less than twice its length. If the area of the rectangle is 162 , what is the length of the diagonal?

Answer 1

Area = length * width

162 = x ( 2x-1)

162 = 2x^2 - x

0 = 2x^2 -x - 162

per quadratic formula the only valid solution is x=9.25347

the width is then 17.50694

the diagonal per pythagorean is 19.80201139289895469784385528676

Answer 2

W=2L-1

WL=162=(2L-1)L

2L² - L - 162 = 0

This quadratic has only one solution which makes sense and by quadratic formula it is

L = 9.253472.

Then W = 18.506943

The diagonal will be sqrt(L² + W²).

I always worry when problems give ugly results like this, but I believe this to be correct.