Solve alphax^2+x+1=0

$$\alpha x ^ { 2 } + x + 1 = 0$$

Solve for α

$$\alpha =-\frac{x+1}{x^{2}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

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

$$\alpha x^{2}+1=-x$$

Subtract 1 from both sides.

$$\alpha x^{2}=-x-1$$

The equation is in standard form.

$$x^{2}\alpha =-x-1$$

Divide both sides by x^{2}.

$$\frac{x^{2}\alpha }{x^{2}}=\frac{-x-1}{x^{2}}$$

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

$$\alpha =\frac{-x-1}{x^{2}}$$

Divide -x-1 by x^{2}.

$$\alpha =-\frac{x+1}{x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{1-4\alpha }-1}{2\alpha }\text{; }x=-\frac{\sqrt{1-4\alpha }+1}{2\alpha }\text{, }&\alpha \neq 0\\x=-1\text{, }&\alpha =0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{1-4\alpha }-1}{2\alpha }\text{; }x=-\frac{\sqrt{1-4\alpha }+1}{2\alpha }\text{, }&\alpha \neq 0\text{ and }\alpha \leq \frac{1}{4}\\x=-1\text{, }&\alpha =0\end{matrix}\right.$$