Solve frac{sqrt{2}x+4x^2-2y}{2y}=0

$$\frac { \sqrt { 2 } x + 4 x ^ { 2 } - 2 y } { 2 y } = 0$$

Solve for y

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

$$x\neq -\frac{\sqrt{2}}{4}\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2y.

$$\sqrt{2}x+4x^{2}-2y=0$$

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

$$4x^{2}-2y=-\sqrt{2}x$$

Subtract 4x^{2} from both sides.

$$-2y=-\sqrt{2}x-4x^{2}$$

Reorder the terms.

$$-2y=-4x^{2}-\sqrt{2}x$$

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide -x\left(4x+\sqrt{2}\right) by -2.

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

Variable y cannot be equal to 0.

$$y=2x^{2}+\frac{\sqrt{2}x}{2}\text{, }y\neq 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{2}\left(\sqrt{16y+1}-1\right)}{8}$$

$$x=-\frac{\sqrt{2}\left(\sqrt{16y+1}+1\right)}{8}\text{, }y\neq 0$$

Solve for x

$$x=\frac{\sqrt{2}\left(\sqrt{16y+1}-1\right)}{8}$$

$$x=-\frac{\sqrt{2}\left(\sqrt{16y+1}+1\right)}{8}\text{, }y\geq -\frac{1}{16}\text{ and }y\neq 0$$