Solve 4x+sqrt{8}xy^2=25

$$4 x + \sqrt { 8 } x y ^ { 2 } = 25$$

Solve for x (complex solution)

$$x=\frac{25\sqrt{2}}{4\left(y^{2}+\sqrt{2}\right)}$$

$$y\neq -\sqrt[4]{2}i\text{ and }y\neq \sqrt[4]{2}i$$

Steps for Solving Linear Equation

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$4x+2\sqrt{2}xy^{2}=25$$

Combine all terms containing x.

$$\left(4+2\sqrt{2}y^{2}\right)x=25$$

The equation is in standard form.

$$\left(2\sqrt{2}y^{2}+4\right)x=25$$

Divide both sides by 4+2\sqrt{2}y^{2}.

$$\frac{\left(2\sqrt{2}y^{2}+4\right)x}{2\sqrt{2}y^{2}+4}=\frac{25}{2\sqrt{2}y^{2}+4}$$

Dividing by 4+2\sqrt{2}y^{2} undoes the multiplication by 4+2\sqrt{2}y^{2}.

$$x=\frac{25}{2\sqrt{2}y^{2}+4}$$

Divide 25 by 4+2\sqrt{2}y^{2}.

$$x=\frac{25\sqrt{2}}{4\left(y^{2}+\sqrt{2}\right)}$$

Solve for x

$$x=\frac{25\sqrt{2}}{4\left(y^{2}+\sqrt{2}\right)}$$

Steps for Solving Linear Equation

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$4x+2\sqrt{2}xy^{2}=25$$

Combine all terms containing x.

$$\left(4+2\sqrt{2}y^{2}\right)x=25$$

The equation is in standard form.

$$\left(2\sqrt{2}y^{2}+4\right)x=25$$

Divide both sides by 4+2\sqrt{2}y^{2}.

$$\frac{\left(2\sqrt{2}y^{2}+4\right)x}{2\sqrt{2}y^{2}+4}=\frac{25}{2\sqrt{2}y^{2}+4}$$

Dividing by 4+2\sqrt{2}y^{2} undoes the multiplication by 4+2\sqrt{2}y^{2}.

$$x=\frac{25}{2\sqrt{2}y^{2}+4}$$

Divide 25 by 4+2\sqrt{2}y^{2}.

$$x=\frac{25\sqrt{2}}{4\left(y^{2}+\sqrt{2}\right)}$$

Solve for y (complex solution)

$$y=-\frac{\sqrt[4]{2}ix^{-\frac{1}{2}}\sqrt{4x-25}}{2}$$

$$y=\frac{\sqrt[4]{2}ix^{-\frac{1}{2}}\sqrt{4x-25}}{2}\text{, }x\neq 0$$

Solve for y

$$y=\frac{\sqrt[4]{2}\sqrt{-4+\frac{25}{x}}}{2}$$

$$y=-\frac{\sqrt[4]{2}\sqrt{-4+\frac{25}{x}}}{2}\text{, }x>0\text{ and }x\leq \frac{25}{4}$$