Solve ax^2+bx^3+cx=0

$$a x ^ { 2 } + b x ^ { 3 } + c x = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-bx-\frac{c}{x}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract bx^{3} from both sides. Anything subtracted from zero gives its negation.

$$ax^{2}+cx=-bx^{3}$$

Subtract cx from both sides.

$$ax^{2}=-bx^{3}-cx$$

The equation is in standard form.

$$x^{2}a=-bx^{3}-cx$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=-\frac{x\left(bx^{2}+c\right)}{x^{2}}$$

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

$$a=-\frac{x\left(bx^{2}+c\right)}{x^{2}}$$

Divide -x\left(bx^{2}+c\right) by x^{2}.

$$a=-bx-\frac{c}{x}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{ax+c}{x^{2}}\text{, }&x\neq 0\\b\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ax^{2} from both sides. Anything subtracted from zero gives its negation.

$$bx^{3}+cx=-ax^{2}$$

Subtract cx from both sides.

$$bx^{3}=-ax^{2}-cx$$

The equation is in standard form.

$$x^{3}b=-ax^{2}-cx$$

Divide both sides by x^{3}.

$$\frac{x^{3}b}{x^{3}}=-\frac{x\left(ax+c\right)}{x^{3}}$$

Dividing by x^{3} undoes the multiplication by x^{3}.

$$b=-\frac{x\left(ax+c\right)}{x^{3}}$$

Divide -x\left(ax+c\right) by x^{3}.

$$b=-\frac{ax+c}{x^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=-bx-\frac{c}{x}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract bx^{3} from both sides. Anything subtracted from zero gives its negation.

$$ax^{2}+cx=-bx^{3}$$

Subtract cx from both sides.

$$ax^{2}=-bx^{3}-cx$$

The equation is in standard form.

$$x^{2}a=-bx^{3}-cx$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=-\frac{x\left(bx^{2}+c\right)}{x^{2}}$$

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

$$a=-\frac{x\left(bx^{2}+c\right)}{x^{2}}$$

Divide -x\left(bx^{2}+c\right) by x^{2}.

$$a=-bx-\frac{c}{x}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax+c}{x^{2}}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ax^{2} from both sides. Anything subtracted from zero gives its negation.

$$bx^{3}+cx=-ax^{2}$$

Subtract cx from both sides.

$$bx^{3}=-ax^{2}-cx$$

The equation is in standard form.

$$x^{3}b=-ax^{2}-cx$$

Divide both sides by x^{3}.

$$\frac{x^{3}b}{x^{3}}=-\frac{x\left(ax+c\right)}{x^{3}}$$

Dividing by x^{3} undoes the multiplication by x^{3}.

$$b=-\frac{x\left(ax+c\right)}{x^{3}}$$

Divide -x\left(ax+c\right) by x^{3}.

$$b=-\frac{ax+c}{x^{2}}$$