Solve y^2=4beta^2(beta^2-x)

$$y ^ { 2 } = 4 \beta ^ { 2 } ( \beta ^ { 2 } - x )$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\beta ^{2}-\frac{\left(\frac{y}{\beta }\right)^{2}}{4}\text{, }&\beta \neq 0\\x\in \mathrm{C}\text{, }&y=0\text{ and }\beta =0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 4\beta ^{2} by \beta ^{2}-x.

$$y^{2}=4\beta ^{4}-4\beta ^{2}x$$

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

$$4\beta ^{4}-4\beta ^{2}x=y^{2}$$

Subtract 4\beta ^{4} from both sides.

$$-4\beta ^{2}x=y^{2}-4\beta ^{4}$$

The equation is in standard form.

$$\left(-4\beta ^{2}\right)x=y^{2}-4\beta ^{4}$$

Divide both sides by -4\beta ^{2}.

$$\frac{\left(-4\beta ^{2}\right)x}{-4\beta ^{2}}=\frac{y^{2}-4\beta ^{4}}{-4\beta ^{2}}$$

Dividing by -4\beta ^{2} undoes the multiplication by -4\beta ^{2}.

$$x=\frac{y^{2}-4\beta ^{4}}{-4\beta ^{2}}$$

Divide y^{2}-4\beta ^{4} by -4\beta ^{2}.

$$x=\beta ^{2}-\frac{y^{2}}{4\beta ^{2}}$$

Solve for x

$$\left\{\begin{matrix}x=\beta ^{2}-\frac{\left(\frac{y}{\beta }\right)^{2}}{4}\text{, }&\beta \neq 0\\x\in \mathrm{R}\text{, }&y=0\text{ and }\beta =0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 4\beta ^{2} by \beta ^{2}-x.

$$y^{2}=4\beta ^{4}-4\beta ^{2}x$$

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

$$4\beta ^{4}-4\beta ^{2}x=y^{2}$$

Subtract 4\beta ^{4} from both sides.

$$-4\beta ^{2}x=y^{2}-4\beta ^{4}$$

The equation is in standard form.

$$\left(-4\beta ^{2}\right)x=y^{2}-4\beta ^{4}$$

Divide both sides by -4\beta ^{2}.

$$\frac{\left(-4\beta ^{2}\right)x}{-4\beta ^{2}}=\frac{y^{2}-4\beta ^{4}}{-4\beta ^{2}}$$

Dividing by -4\beta ^{2} undoes the multiplication by -4\beta ^{2}.

$$x=\frac{y^{2}-4\beta ^{4}}{-4\beta ^{2}}$$

Divide y^{2}-4\beta ^{4} by -4\beta ^{2}.

$$x=\beta ^{2}-\frac{y^{2}}{4\beta ^{2}}$$

Solve for y (complex solution)

$$y=-2\beta \sqrt{\beta ^{2}-x}$$

$$y=2\beta \sqrt{\beta ^{2}-x}$$

Solve for y

$$\left\{\begin{matrix}y=2\beta \sqrt{\beta ^{2}-x}\text{; }y=-2\beta \sqrt{\beta ^{2}-x}\text{, }&x\leq \beta ^{2}\\y=0\text{, }&\beta =0\end{matrix}\right.$$