Solve ax^2+b/ax+c/a=0

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

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Multiply both sides of the equation by a.

$$ax^{2}a+bx+c=0$$

Multiply a and a to get a^{2}.

$$a^{2}x^{2}+bx+c=0$$

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

$$bx+c=-a^{2}x^{2}$$

Subtract c from both sides.

$$bx=-a^{2}x^{2}-c$$

The equation is in standard form.

$$xb=-a^{2}x^{2}-c$$

Divide both sides by x.

$$\frac{xb}{x}=\frac{-a^{2}x^{2}-c}{x}$$

Dividing by x undoes the multiplication by x.

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

Divide -a^{2}x^{2}-c by x.

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

Solve for b

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

Steps for Solving Linear Equation

Multiply both sides of the equation by a.

$$ax^{2}a+bx+c=0$$

Multiply a and a to get a^{2}.

$$a^{2}x^{2}+bx+c=0$$

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

$$bx+c=-a^{2}x^{2}$$

Subtract c from both sides.

$$bx=-a^{2}x^{2}-c$$

The equation is in standard form.

$$xb=-a^{2}x^{2}-c$$

Divide both sides by x.

$$\frac{xb}{x}=\frac{-a^{2}x^{2}-c}{x}$$

Dividing by x undoes the multiplication by x.

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

Divide -a^{2}x^{2}-c by x.

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{i\sqrt{bx+c}}{x}\text{; }a=\frac{i\sqrt{bx+c}}{x}\text{, }&\left(c\neq 0\text{ or }b\neq 0\right)\text{ and }\left(b=0\text{ or }x\neq -\frac{c}{b}\right)\text{ and }x\neq 0\text{ and }c\neq -bx\\a\neq 0\text{, }&c=0\text{ and }x=0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=\frac{\sqrt{-bx-c}}{|x|}\text{; }a=-\frac{\sqrt{-bx-c}}{|x|}\text{, }&\left(b=0\text{ and }x\neq 0\text{ and }c<0\right)\text{ or }\left(x\neq 0\text{ and }x<-\frac{c}{b}\text{ and }c<-bx\text{ and }b>0\right)\text{ or }\left(x\neq 0\text{ and }x>-\frac{c}{b}\text{ and }c<-bx\text{ and }b<0\right)\\a\neq 0\text{, }&c=0\text{ and }x=0\end{matrix}\right.$$