Solutions of $0 = A\sin^2(\alpha) + B\sin(2\alpha) - C$? I am facing a problem from physics class involving a projectile motion which can be described with such an given equation:
$$
h = -\frac{1}{2} \frac{g}{v_{0}^2 \cos^2\alpha} d^2 + \frac{\sin\alpha}{\cos\alpha} d + y.
$$
The goal is to find the minimum value of (rearranging above equation)
$$
v_0(\alpha) = \frac{d}{\cos\alpha} \cdot \sqrt{\frac{1}{2} \cdot \frac{g}{\tan\alpha \cdot d + y - h}}.
$$
This involves finding solutions to $v_0'(\alpha) = 0$. I was able to find the derivative ($t := d\cdot \tan\alpha +y-h$):
$$
v_0'(\alpha) = \frac{\tan \alpha}{\cos\alpha \cdot \sqrt{t}} - \frac{d}{2\cos^3\alpha \cdot (\sqrt{t})^3}
$$
Because of the condition $v_0'(\alpha) = 0$ this simplifies to (it is know that the solution is around $50^\circ$)
$$
0 = \sin\alpha - \frac{d}{2\sin\alpha \cos\alpha \cdot (d\cdot \tan\alpha + y - h)}
$$
or
$$
0 = d\sin(2\alpha)\tan\alpha + (y-h)\sin(2\alpha) - d.
$$
or
$$
0 = 2d\sin^2(\alpha) + (y-h)\sin(2\alpha) - d
$$
or with some constants

$$ 0 = 2d\sin^2(\alpha) + B\sin(2\alpha) - d. $$ How can I find the
  solutions from here?


You can expand $\sin2\alpha$:
$$
2d\sin^2\alpha+2B\sin\alpha\cos\alpha-d\sin^2\alpha-d\cos^2\alpha=0
$$
that becomes
$$
d\sin^2\alpha+2B\sin\alpha\cos\alpha-d\cos^2\alpha=0
$$
If $d\ne0$, we cannot have $\cos\alpha=0$ as a solution, so we can divide by $\cos^2\alpha$, leading to
$$
d\tan^2\alpha+2B\tan\alpha-d=0
$$
so
$$
\tan\alpha=\frac{-B\pm\sqrt{B^2+d^2}}{d}
$$
