When I use Mathematica to calculate some roots of an quadratic equation, it will output the simplified answer like $\frac{-w \sqrt{-h} \sqrt{4 \theta ^2 p-4 \theta p-h}+h w+2 \theta p w}{2 \theta ( p+h)}$. In this solution, $r, p$ are both required to be nonnegative. So the answer given by Mathematica may result in Imaginary solution. I knew that Mathematica has given an option that "Assumption". But when the solving equation is complex, the computation time is unbearable. So is there another method to get correct result? Or is there some simple method to calculate the product of two squared root terms?
Here is an example:
eqn = (p (p (v - w)^2 \[Theta] + h v (-w + v \[Theta])))/(2 (h + p) v) Solve[eqn == 0, v] The result is $$\left\{\left\{v\to \frac{-\sqrt{-h} w \sqrt{-h+4 \theta ^2 p-4 \theta p}+h w+2 \theta p w}{2 (h \theta +\theta p)}\right\},\left\{v\to \frac{\sqrt{-h} w \sqrt{-h+4 \theta ^2 p-4 \theta p}+h w+2 \theta p w}{2 (h \theta +\theta p)}\right\}\right\}$$
sol = Simplify[Solve[eqn == 0, v], {h >= 0, p >= 0}]$\endgroup$