Solve the quadratic with 4 unknowns

Question

I know how to solve quadratic equations with the quadratic formula but this question has really got me stumped.

$$h=-16t^2+vt+s.$$

I am trying to solve for t but I have no idea where to start since there are 4 variables. Could somebody give a step by step walkthrough to get to a solution for t?

Answer 1

\begin{align} & h = -16t^{2} + vt + s \\ \implies & 16t^{2} - vt + (h -s) = 0 \end{align}

Comparing this with the standard form of $ax^{2} + bx + c = 0$, the roots will be (see here for details)

\begin{equation} t = \dfrac{v \pm \sqrt{(-v)^{2} - 4\times 16 \times(h-s)}}{2\times 16} = \dfrac{v \pm \sqrt{v^{2} - 64(h-s)}}{32} \end{equation}.

Answer 2

writing as $$t^2-\frac{v}{16}t-\frac{s-h}{16}=0$$ and now using the quadratic formula: $$t_{1,2}=\frac{v}{32}\pm\sqrt{\left(\frac{v}{32}\right)^2+\frac{s-h}{16}}$$