Mechanics - Projectile Motion

Problem statement is here: http://www.phys.uri.edu/~gerhard/PHY520/wmex139.pdf

My approach:
[tex]y = -\frac{gt^2}{2} + v_0 cos( \alpha) t + h[/tex]

Projectile hits ground at:
[tex] t_{gnd} = \frac{v_0 sin( \alpha ) + \sqrt{ v_0^2 sin^2 ( \alpha) + 2gh}}{g}[/tex]

Now compute derivative of x and solve for alpha:
[tex]x_{max} = v_0 cos( \alpha) t_{gnd} [/tex]
[tex]\frac{dx_{max}}{d \alpha} = 0[/tex]

This last step where you solve for alpha is what buggers me. I get a huge quartic expression in alpha that does not simplify, I suspect the problem setter expects you to make some approximation but I can't figure it out.

 

Worked like a charm, good idea.