Solve x^3+sqrt{xy}=7

$$x ^ { 3 } + \sqrt { x y } = 7$$

Solve for y

$$y=\frac{\left(7-x^{3}\right)^{2}}{x}$$

$$7-x^{3}\geq 0\text{ and }x\neq 0$$

Steps for Solving Radical Equations

Subtract x^{3} from both sides of the equation.

$$\sqrt{xy}+x^{3}-x^{3}=7-x^{3}$$

Subtracting x^{3} from itself leaves 0.

$$\sqrt{xy}=7-x^{3}$$

Square both sides of the equation.

$$xy=\left(7-x^{3}\right)^{2}$$

Divide both sides by x.

$$\frac{xy}{x}=\frac{\left(7-x^{3}\right)^{2}}{x}$$

Dividing by x undoes the multiplication by x.

$$y=\frac{\left(7-x^{3}\right)^{2}}{x}$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=\frac{\left(7-x^{3}\right)^{2}}{x}\text{, }&x\neq 0\text{ and }arg(7-x^{3})<\pi \\y=0\text{, }&x=\sqrt[3]{7}e^{\frac{4\pi i}{3}}\text{ or }x=\sqrt[3]{7}e^{\frac{2\pi i}{3}}\text{ or }x=\sqrt[3]{7}\end{matrix}\right.$$