Diagonal brace?

Question

The width of a rectangular gate is 2 meters (m) larger than its height. The diagonal brace measures radical 6m. Find the width and height.

Answer 1

w=x+2
h=x
(x+2)^2 +x^2 =6
2x^2+4x+4=6
x^2+2x-1=0
If you use the quadratic equation to solve this, you get [-1+sqrt(2] for the height; the width is just 2 more then this: 1+sqrt(2).

Answer 2

The width, height, and brace, form a right triangle. the width and length are 2 sides and the brace is the hypotenuse.

you're given that w=2+h.. using this and pyhagorean's theorem you can find the other side

w=2+h
b = sqrt(6)

(2+h)^2 + h^2 = 6
h^2+4h+4+h^2 = 6
2h^2+4h-2 =0
h^2 + 2h -1 = 0

use quadratic formula or to find solutions
-2 + sqrt(4+4)/2 = sqrt(2) -1
-2 - sqrt(4+4)/2 = -sqrt(2)-1

discard the negative solution
h = sqrt(2) -1
w = sqrt(2) - 1 +2
w = sqrt(2) + 1

h = sqrt(2) -1
w = sqrt(2) +1

you can use Pythagorean theorem again to see that it works