Solve x^2+(-2+sqrt{5})y=10+3sqrt{5}

$$x ^ { 2 } + ( - 2 + \sqrt { 5 } ) y = 10 + 3 \sqrt { 5 }$$

Solve for y

$$y=-\left(\sqrt{5}+2\right)\left(x^{2}-3\sqrt{5}-10\right)$$

Steps for Solving Linear Equation

Use the distributive property to multiply -2+\sqrt{5} by y.

$$x^{2}-2y+\sqrt{5}y=10+3\sqrt{5}$$

Subtract x^{2} from both sides.

$$-2y+\sqrt{5}y=10+3\sqrt{5}-x^{2}$$

Combine all terms containing y.

$$\left(-2+\sqrt{5}\right)y=10+3\sqrt{5}-x^{2}$$

The equation is in standard form.

$$\left(\sqrt{5}-2\right)y=-x^{2}+3\sqrt{5}+10$$

Divide both sides by -2+\sqrt{5}.

$$\frac{\left(\sqrt{5}-2\right)y}{\sqrt{5}-2}=\frac{-x^{2}+3\sqrt{5}+10}{\sqrt{5}-2}$$

Dividing by -2+\sqrt{5} undoes the multiplication by -2+\sqrt{5}.

$$y=\frac{-x^{2}+3\sqrt{5}+10}{\sqrt{5}-2}$$

Divide 10+3\sqrt{5}-x^{2} by -2+\sqrt{5}.

$$y=\left(\sqrt{5}+2\right)\left(-x^{2}+3\sqrt{5}+10\right)$$

Solve for x (complex solution)

$$x=-\sqrt{-\sqrt{5}y+2y+3\sqrt{5}+10}$$

$$x=\sqrt{-\sqrt{5}y+2y+3\sqrt{5}+10}$$

Solve for x

$$x=\sqrt{-\sqrt{5}y+2y+3\sqrt{5}+10}$$

$$x=-\sqrt{-\sqrt{5}y+2y+3\sqrt{5}+10}\text{, }y\leq 16\sqrt{5}+35$$