Solve zt^2+9t+4=0

$$z t ^ { 2 } + 9 t + 4 = 0$$

Solve for z

$$z=-\frac{9t+4}{t^{2}}$$

$$t\neq 0$$

Steps for Solving Linear Equation

Subtract 9t from both sides. Anything subtracted from zero gives its negation.

$$zt^{2}+4=-9t$$

Subtract 4 from both sides.

$$zt^{2}=-9t-4$$

The equation is in standard form.

$$t^{2}z=-9t-4$$

Divide both sides by t^{2}.

$$\frac{t^{2}z}{t^{2}}=\frac{-9t-4}{t^{2}}$$

Dividing by t^{2} undoes the multiplication by t^{2}.

$$z=\frac{-9t-4}{t^{2}}$$

Divide -9t-4 by t^{2}.

$$z=-\frac{9t+4}{t^{2}}$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=\frac{\sqrt{81-16z}-9}{2z}\text{; }t=-\frac{\sqrt{81-16z}+9}{2z}\text{, }&z\neq 0\\t=-\frac{4}{9}\approx -0.444444444\text{, }&z=0\end{matrix}\right.$$

Solve for t

$$\left\{\begin{matrix}t=\frac{\sqrt{81-16z}-9}{2z}\text{; }t=-\frac{\sqrt{81-16z}+9}{2z}\text{, }&z\neq 0\text{ and }z\leq \frac{81}{16}\\t=-\frac{4}{9}\approx -0.444444444\text{, }&z=0\end{matrix}\right.$$