Solve -2x^2+5sqrt{2}x=(4-sqrt{2})y+sqrt{2}

$$-2 { x }^{ 2 } +5 \sqrt{ 2 } x=(4- \sqrt{ 2 } )y+ \sqrt{ 2 }$$

Solve for y

$$y=-\frac{\left(2\sqrt{2}+1\right)\left(\sqrt{2}x^{2}-5x+1\right)}{7}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 4-\sqrt{2} by y.

$$-2x^{2}+5\sqrt{2}x=4y-\sqrt{2}y+\sqrt{2}$$

Swap sides so that all variable terms are on the left hand side.

$$4y-\sqrt{2}y+\sqrt{2}=-2x^{2}+5\sqrt{2}x$$

Subtract \sqrt{2} from both sides.

$$4y-\sqrt{2}y=-2x^{2}+5\sqrt{2}x-\sqrt{2}$$

Combine all terms containing y.

$$\left(4-\sqrt{2}\right)y=-2x^{2}+5\sqrt{2}x-\sqrt{2}$$

Divide both sides by 4-\sqrt{2}.

$$\frac{\left(4-\sqrt{2}\right)y}{4-\sqrt{2}}=\frac{-2x^{2}+5\sqrt{2}x-\sqrt{2}}{4-\sqrt{2}}$$

Dividing by 4-\sqrt{2} undoes the multiplication by 4-\sqrt{2}.

$$y=\frac{-2x^{2}+5\sqrt{2}x-\sqrt{2}}{4-\sqrt{2}}$$

Divide -2x^{2}+5\sqrt{2}x-\sqrt{2} by 4-\sqrt{2}.

$$y=\frac{\left(\sqrt{2}+4\right)\left(-2x^{2}+5\sqrt{2}x-\sqrt{2}\right)}{14}$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{8\sqrt{2}y-32y+50-8\sqrt{2}}+5\sqrt{2}}{4}$$

$$x=\frac{\sqrt{8\sqrt{2}y-32y+50-8\sqrt{2}}+5\sqrt{2}}{4}$$

Solve for x

$$x=\frac{-\sqrt{8\sqrt{2}y-32y+50-8\sqrt{2}}+5\sqrt{2}}{4}$$

$$x=\frac{\sqrt{8\sqrt{2}y-32y+50-8\sqrt{2}}+5\sqrt{2}}{4}\text{, }y\leq \frac{9\sqrt{2}}{56}+\frac{23}{14}$$