Finding the length of a diagonal

Question

The width of a rectangle is two less than twice its length if the area of the rectangle is 26 cm² what is the length of the diagonal?

Answer 1

Area = 1/2 * base * height

26 = (1/2) * x ( 2x-2)

26 = (1/2)*2 * x (x-1)

26 = x(x-1)

x^2 -x - 26 = 0

x = (1 +sqrt ( 1 + 4*26))/2 <-- negative branch disqualified

= (1 + sqrt( 105))/2 <-- this is the base

= (1 + sqrt(5*7*3))/2

2x-2 = sqrt(105)-1 <-- this is the height

area = (1/2) [sqrt(105)-1] (1/2) [ sqrt(105)+1]

(1/4) [ 105 - 1] = 1/4*104 = 26

so yes it checks