Solve ex^2=-x-3by

$$e x ^ { 2 } = - x - 3 b y$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{x\left(ex+1\right)}{3y}\text{, }&y\neq 0\\b\in \mathrm{C}\text{, }&\left(x=0\text{ or }x=-\frac{1}{e}\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$-x-3by=ex^{2}$$

Add x to both sides.

$$-3by=ex^{2}+x$$

The equation is in standard form.

$$\left(-3y\right)b=ex^{2}+x$$

Divide both sides by -3y.

$$\frac{\left(-3y\right)b}{-3y}=\frac{x\left(ex+1\right)}{-3y}$$

Dividing by -3y undoes the multiplication by -3y.

$$b=\frac{x\left(ex+1\right)}{-3y}$$

Divide x\left(ex+1\right) by -3y.

$$b=-\frac{x\left(ex+1\right)}{3y}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{x\left(ex+1\right)}{3y}\text{, }&y\neq 0\\b\in \mathrm{R}\text{, }&\left(x=0\text{ or }x=-\frac{1}{e}\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$-x-3by=ex^{2}$$

Add x to both sides.

$$-3by=ex^{2}+x$$

The equation is in standard form.

$$\left(-3y\right)b=ex^{2}+x$$

Divide both sides by -3y.

$$\frac{\left(-3y\right)b}{-3y}=\frac{x\left(ex+1\right)}{-3y}$$

Dividing by -3y undoes the multiplication by -3y.

$$b=\frac{x\left(ex+1\right)}{-3y}$$

Divide x\left(ex+1\right) by -3y.

$$b=-\frac{x\left(ex+1\right)}{3y}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{1-12eby}-1}{2e}$$

$$x=-\frac{\sqrt{1-12eby}+1}{2e}$$

Solve for x

$$x=\frac{\sqrt{1-12eby}-1}{2e}$$

$$x=-\frac{\sqrt{1-12eby}+1}{2e}\text{, }\left(y\geq 0\text{ or }b\geq \frac{1}{12ey}\right)\text{ and }\left(y\leq 0\text{ or }b\leq \frac{1}{12ey}\right)$$