Solve alphax^2+betax+gamma=0

$$\alpha x ^ { 2 } + \beta x + \gamma = 0$$

Solve for α (complex solution)

$$\left\{\begin{matrix}\alpha =-\frac{x\beta +\gamma }{x^{2}}\text{, }&x\neq 0\\\alpha \in \mathrm{C}\text{, }&\gamma =0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \beta x from both sides. Anything subtracted from zero gives its negation.

$$\alpha x^{2}+\gamma =-\beta x$$

Subtract \gamma from both sides.

$$\alpha x^{2}=-\beta x-\gamma$$

Reorder the terms.

$$\alpha x^{2}=-x\beta -\gamma$$

The equation is in standard form.

$$x^{2}\alpha =-x\beta -\gamma$$

Divide both sides by x^{2}.

$$\frac{x^{2}\alpha }{x^{2}}=\frac{-x\beta -\gamma }{x^{2}}$$

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

$$\alpha =\frac{-x\beta -\gamma }{x^{2}}$$

Divide -x\beta -\gamma by x^{2}.

$$\alpha =-\frac{x\beta +\gamma }{x^{2}}$$

Solve for α

$$\left\{\begin{matrix}\alpha =-\frac{x\beta +\gamma }{x^{2}}\text{, }&x\neq 0\\\alpha \in \mathrm{R}\text{, }&\gamma =0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \beta x from both sides. Anything subtracted from zero gives its negation.

$$\alpha x^{2}+\gamma =-\beta x$$

Subtract \gamma from both sides.

$$\alpha x^{2}=-\beta x-\gamma$$

Reorder the terms.

$$\alpha x^{2}=-x\beta -\gamma$$

The equation is in standard form.

$$x^{2}\alpha =-x\beta -\gamma$$

Divide both sides by x^{2}.

$$\frac{x^{2}\alpha }{x^{2}}=\frac{-x\beta -\gamma }{x^{2}}$$

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

$$\alpha =\frac{-x\beta -\gamma }{x^{2}}$$

Divide -x\beta -\gamma by x^{2}.

$$\alpha =-\frac{x\beta +\gamma }{x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{\beta ^{2}-4\alpha \gamma }-\beta }{2\alpha }\text{; }x=-\frac{\sqrt{\beta ^{2}-4\alpha \gamma }+\beta }{2\alpha }\text{, }&\alpha \neq 0\\x=-\frac{\gamma }{\beta }\text{, }&\alpha =0\text{ and }\beta \neq 0\\x\in \mathrm{C}\text{, }&\alpha =0\text{ and }\beta =0\text{ and }\gamma =0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{\beta ^{2}-4\alpha \gamma }-\beta }{2\alpha }\text{; }x=-\frac{\sqrt{\beta ^{2}-4\alpha \gamma }+\beta }{2\alpha }\text{, }&\left(\beta \neq 0\text{ and }\alpha =\frac{\beta ^{2}}{4\gamma }\text{ and }\gamma \neq 0\right)\text{ or }\left(\alpha \neq 0\text{ and }\alpha \geq \frac{\beta ^{2}}{4\gamma }\text{ and }\gamma \leq 0\right)\text{ or }\left(\gamma =0\text{ and }\alpha \neq 0\right)\text{ or }\left(\alpha \neq 0\text{ and }\gamma \geq 0\text{ and }\alpha \leq \frac{\beta ^{2}}{4\gamma }\right)\\x=-\frac{\gamma }{\beta }\text{, }&\alpha =0\text{ and }\beta \neq 0\\x\in \mathrm{R}\text{, }&\alpha =0\text{ and }\beta =0\text{ and }\gamma =0\end{matrix}\right.$$