Solve 2d^2x/dt^2+3dx/dt+2=0

$$\frac { 2 d ^ { 2 } x } { d t ^ { 2 } } + \frac { 3 d x } { d t } + 2 = 0$$

Solve for t (complex solution)

$$t=\frac{\sqrt{x\left(9x-16d\right)}-3x}{4}$$

$$t=\frac{-\sqrt{x\left(9x-16d\right)}-3x}{4}\text{, }d\neq 0\text{ and }x\neq 0$$

Solve for d

$$d=-\frac{t^{2}}{x}-\frac{3t}{2}$$

$$t\neq 0\text{ and }x\neq -\frac{2t}{3}\text{ and }x\neq 0$$

Solve for t

$$t=\frac{\sqrt{x\left(9x-16d\right)}-3x}{4}$$

$$t=\frac{-\sqrt{x\left(9x-16d\right)}-3x}{4}\text{, }\left(x>0\text{ and }d\leq \frac{9x}{16}\text{ and }d\neq 0\right)\text{ or }\left(x\neq 0\text{ and }d=\frac{9x}{16}\right)\text{ or }\left(d\neq 0\text{ and }d\geq \frac{9x}{16}\text{ and }x<0\right)$$