Solve x^2+3x(x^2-y)=0

$$x ^ { 2 } + 3 x ( x ^ { 2 } - y ) = 0$$

Solve for y (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply 3x by x^{2}-y.

$$x^{2}+3x^{3}-3xy=0$$

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

$$3x^{3}-3xy=-x^{2}$$

Subtract 3x^{3} from both sides.

$$-3xy=-x^{2}-3x^{3}$$

The equation is in standard form.

$$\left(-3x\right)y=-3x^{3}-x^{2}$$

Divide both sides by -3x.

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

Dividing by -3x undoes the multiplication by -3x.

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

Divide -\left(1+3x\right)x^{2} by -3x.

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

Solve for y

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

Steps for Solving Linear Equation

Use the distributive property to multiply 3x by x^{2}-y.

$$x^{2}+3x^{3}-3xy=0$$

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

$$3x^{3}-3xy=-x^{2}$$

Subtract 3x^{3} from both sides.

$$-3xy=-x^{2}-3x^{3}$$

The equation is in standard form.

$$\left(-3x\right)y=-3x^{3}-x^{2}$$

Divide both sides by -3x.

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

Dividing by -3x undoes the multiplication by -3x.

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

Divide -\left(1+3x\right)x^{2} by -3x.

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

Solve for x (complex solution)

$$x=\frac{-\sqrt{36y+1}-1}{6}$$

$$x=0$$

$$x=\frac{\sqrt{36y+1}-1}{6}$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{36y+1}-1}{6}\text{; }x=\frac{-\sqrt{36y+1}-1}{6}\text{, }&y\geq -\frac{1}{36}\end{matrix}\right.$$