Solve y^2=ax^3

$$y ^ { 2 } = a x ^ { 3 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{y^{2}}{x^{3}}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax^{3}=y^{2}$$

The equation is in standard form.

$$x^{3}a=y^{2}$$

Divide both sides by x^{3}.

$$\frac{x^{3}a}{x^{3}}=\frac{y^{2}}{x^{3}}$$

Dividing by x^{3} undoes the multiplication by x^{3}.

$$a=\frac{y^{2}}{x^{3}}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{y^{2}}{x^{3}}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax^{3}=y^{2}$$

The equation is in standard form.

$$x^{3}a=y^{2}$$

Divide both sides by x^{3}.

$$\frac{x^{3}a}{x^{3}}=\frac{y^{2}}{x^{3}}$$

Dividing by x^{3} undoes the multiplication by x^{3}.

$$a=\frac{y^{2}}{x^{3}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=e^{\frac{2\pi i}{3}}a^{-\frac{1}{3}}y^{\frac{2}{3}}\text{; }x=a^{-\frac{1}{3}}y^{\frac{2}{3}}\text{; }x=e^{\frac{4\pi i}{3}}a^{-\frac{1}{3}}y^{\frac{2}{3}}\text{, }&a\neq 0\\x\in \mathrm{C}\text{, }&y=0\text{ and }a=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{y^{\frac{2}{3}}}{\sqrt[3]{a}}\text{, }&a\neq 0\\x\in \mathrm{R}\text{, }&y=0\text{ and }a=0\end{matrix}\right.$$