Solve 24sqrt{y}=93-5x^2

$$24 \sqrt { y } = 93 - 5 x ^ { 2 }$$

Solve for y

$$y=\frac{\left(93-5x^{2}\right)^{2}}{576}$$

$$-\frac{5x^{2}}{24}+\frac{31}{8}\geq 0$$

Steps for Solving Radical Equations

Divide both sides by 24.

$$\frac{24\sqrt{y}}{24}=\frac{93-5x^{2}}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$\sqrt{y}=\frac{93-5x^{2}}{24}$$

Divide 93-5x^{2} by 24.

$$\sqrt{y}=-\frac{5x^{2}}{24}+\frac{31}{8}$$

Square both sides of the equation.

$$y=\frac{\left(93-5x^{2}\right)^{2}}{576}$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{-120\sqrt{y}+465}}{5}$$

$$x=\frac{\sqrt{-120\sqrt{y}+465}}{5}$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=\frac{\left(93-5x^{2}\right)^{2}}{576}\text{, }&arg(-\frac{5x^{2}}{24}+\frac{31}{8})<\pi \\y=0\text{, }&x=-\frac{\sqrt{465}}{5}\text{ or }x=\frac{\sqrt{465}}{5}\end{matrix}\right.$$

Solve for x

$$x=\frac{\sqrt{-120\sqrt{y}+465}}{5}$$

$$x=-\frac{\sqrt{-120\sqrt{y}+465}}{5}\text{, }y\geq 0\text{ and }y\leq \frac{961}{64}$$