Solve x^2-x+1-3alphabeta=0

$$x ^ { 2 } - x + 1 - 3 \alpha \beta = 0$$

Solve for α (complex solution)

$$\left\{\begin{matrix}\alpha =\frac{x^{2}-x+1}{3\beta }\text{, }&\beta \neq 0\\\alpha \in \mathrm{C}\text{, }&\left(x=\frac{-\sqrt{3}i+1}{2}\text{ or }x=\frac{1+\sqrt{3}i}{2}\right)\text{ and }\beta =0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$-x+1-3\alpha \beta =-x^{2}$$

Add x to both sides.

$$1-3\alpha \beta =-x^{2}+x$$

Subtract 1 from both sides.

$$-3\alpha \beta =-x^{2}+x-1$$

The equation is in standard form.

$$\left(-3\beta \right)\alpha =-x^{2}+x-1$$

Divide both sides by -3\beta .

$$\frac{\left(-3\beta \right)\alpha }{-3\beta }=\frac{-x^{2}+x-1}{-3\beta }$$

Dividing by -3\beta undoes the multiplication by -3\beta .

$$\alpha =\frac{-x^{2}+x-1}{-3\beta }$$

Divide -x^{2}+x-1 by -3\beta .

$$\alpha =-\frac{-x^{2}+x-1}{3\beta }$$

Solve for α

$$\alpha =\frac{x^{2}-x+1}{3\beta }$$

$$\beta \neq 0$$

Steps for Solving Linear Equation

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

$$-x+1-3\alpha \beta =-x^{2}$$

Add x to both sides.

$$1-3\alpha \beta =-x^{2}+x$$

Subtract 1 from both sides.

$$-3\alpha \beta =-x^{2}+x-1$$

The equation is in standard form.

$$\left(-3\beta \right)\alpha =-x^{2}+x-1$$

Divide both sides by -3\beta .

$$\frac{\left(-3\beta \right)\alpha }{-3\beta }=\frac{-x^{2}+x-1}{-3\beta }$$

Dividing by -3\beta undoes the multiplication by -3\beta .

$$\alpha =\frac{-x^{2}+x-1}{-3\beta }$$

Divide -x^{2}+x-1 by -3\beta .

$$\alpha =-\frac{-x^{2}+x-1}{3\beta }$$

Solve for x (complex solution)

$$x=\frac{\sqrt{12\alpha \beta -3}+1}{2}$$

$$x=\frac{-\sqrt{12\alpha \beta -3}+1}{2}$$

Solve for x

$$x=\frac{\sqrt{12\alpha \beta -3}+1}{2}$$

$$x=\frac{-\sqrt{12\alpha \beta -3}+1}{2}\text{, }\left(\beta <0\text{ or }\alpha \geq \frac{1}{4\beta }\right)\text{ and }\left(\beta >0\text{ or }\alpha \leq \frac{1}{4\beta }\right)\text{ and }\beta \neq 0$$