I'm currently working on a solution to a projectile motion problem, and I need to figure out the optimal launch angles by solving this system of equations.
$15\cos(\theta)x+2\cos(\theta)=21 \\\\ (15\sin(\theta)-\frac{1}{2}9.81x)x+2\sin(\theta)+3=0$
The first equation models the horizontal position of the projectile (which must be $21$ meters to the right in order to hit the target), and the second equation models the projectile's vertical height (which must be $0$ meters from the ground in order to hit the target). My goal was to solve for $x$ (time in seconds), and then plug those values into the original system to solve for $\theta$ (launch angle in degrees). After doing some simplification and substitution, I ended up with this equation:
$24.059025x^4-254.43x^2-60x+446=0$
I know that one approach to solving for $x$ would be to graph the function to see its zeroes. However, is it possible to solve this without graphing it?
