Solve g^2=g[gx]

$$g ^ { 2 } = g [ g x ]$$

Solve for g (complex solution)

$$\left\{\begin{matrix}\\g=0\text{, }&\text{unconditionally}\\g\in \mathrm{C}\text{, }&x=1\end{matrix}\right.$$

Steps by Finding Square Root

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

$$g^{2}=g^{2}x$$

Subtract g^{2}x from both sides.

$$g^{2}-g^{2}x=0$$

Reorder the terms.

$$-xg^{2}+g^{2}=0$$

Combine all terms containing g.

$$\left(-x+1\right)g^{2}=0$$

Dividing by 1-x undoes the multiplication by 1-x.

$$g^{2}=\frac{0}{1-x}$$

Divide 0 by 1-x.

$$g^{2}=0$$

Take the square root of both sides of the equation.

$$g=0 g=0$$

The equation is now solved. Solutions are the same.

$$g=0$$

Steps Using the Quadratic Formula

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

$$g^{2}=g^{2}x$$

Subtract g^{2}x from both sides.

$$g^{2}-g^{2}x=0$$

Reorder the terms.

$$-xg^{2}+g^{2}=0$$

Combine all terms containing g.

$$\left(-x+1\right)g^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$\left(1-x\right)g^{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1-x for a, 0 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$g=\frac{0±\sqrt{0^{2}}}{2\left(1-x\right)}$$

Take the square root of 0^{2}.

$$g=\frac{0±0}{2\left(1-x\right)}$$

Multiply 2 times 1-x.

$$g=\frac{0}{2-2x}$$

Divide 0 by 2-2x.

$$g=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&g=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$g^{2}=g^{2}x$$

Swap sides so that all variable terms are on the left hand side.

$$g^{2}x=g^{2}$$

Divide both sides by g^{2}.

$$\frac{g^{2}x}{g^{2}}=\frac{g^{2}}{g^{2}}$$

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

$$x=\frac{g^{2}}{g^{2}}$$

Divide g^{2} by g^{2}.

$$x=1$$

Solve for x

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&g=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$g^{2}=g^{2}x$$

Swap sides so that all variable terms are on the left hand side.

$$g^{2}x=g^{2}$$

Divide both sides by g^{2}.

$$\frac{g^{2}x}{g^{2}}=\frac{g^{2}}{g^{2}}$$

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

$$x=\frac{g^{2}}{g^{2}}$$

Divide g^{2} by g^{2}.

$$x=1$$

Solve for g

$$\left\{\begin{matrix}\\g=0\text{, }&\text{unconditionally}\\g\in \mathrm{R}\text{, }&x=1\end{matrix}\right.$$