Solve ysqrt{15-sqrt{176}}=(x-sqrt{44})(sqrt{20+sqrt{396}})

$$y \sqrt { 15 - \sqrt { 176 } } = ( x - \sqrt { 44 } ) ( \sqrt { 20 + \sqrt { 396 } } )$$

Solve for x

$$x=-\frac{5\sqrt{11}y}{2}+\frac{17y}{2}+2\sqrt{11}$$

Steps for Solving Linear Equation

Factor 176=4^{2}\times 11. Rewrite the square root of the product \sqrt{4^{2}\times 11} as the product of square roots \sqrt{4^{2}}\sqrt{11}. Take the square root of 4^{2}.

$$y\sqrt{15-4\sqrt{11}}=\left(x-\sqrt{44}\right)\sqrt{20+\sqrt{396}}$$

Factor 44=2^{2}\times 11. Rewrite the square root of the product \sqrt{2^{2}\times 11} as the product of square roots \sqrt{2^{2}}\sqrt{11}. Take the square root of 2^{2}.

$$y\sqrt{15-4\sqrt{11}}=\left(x-2\sqrt{11}\right)\sqrt{20+\sqrt{396}}$$

Factor 396=6^{2}\times 11. Rewrite the square root of the product \sqrt{6^{2}\times 11} as the product of square roots \sqrt{6^{2}}\sqrt{11}. Take the square root of 6^{2}.

$$y\sqrt{15-4\sqrt{11}}=\left(x-2\sqrt{11}\right)\sqrt{20+6\sqrt{11}}$$

Use the distributive property to multiply x-2\sqrt{11} by \sqrt{20+6\sqrt{11}}.

$$y\sqrt{15-4\sqrt{11}}=x\sqrt{20+6\sqrt{11}}-2\sqrt{11}\sqrt{20+6\sqrt{11}}$$

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

$$x\sqrt{20+6\sqrt{11}}-2\sqrt{11}\sqrt{20+6\sqrt{11}}=y\sqrt{15-4\sqrt{11}}$$

Add 2\sqrt{11}\sqrt{20+6\sqrt{11}} to both sides.

$$x\sqrt{20+6\sqrt{11}}=y\sqrt{15-4\sqrt{11}}+2\sqrt{11}\sqrt{20+6\sqrt{11}}$$

The equation is in standard form.

$$\sqrt{6\sqrt{11}+20}x=\sqrt{15-4\sqrt{11}}y+2\sqrt{11}\sqrt{6\sqrt{11}+20}$$

Divide both sides by \sqrt{20+6\sqrt{11}}.

$$\frac{\sqrt{6\sqrt{11}+20}x}{\sqrt{6\sqrt{11}+20}}=\frac{\left(\sqrt{11}-2\right)y+6\sqrt{11}+22}{\sqrt{6\sqrt{11}+20}}$$

Dividing by \sqrt{20+6\sqrt{11}} undoes the multiplication by \sqrt{20+6\sqrt{11}}.

$$x=\frac{\left(\sqrt{11}-2\right)y+6\sqrt{11}+22}{\sqrt{6\sqrt{11}+20}}$$

Divide y\left(\sqrt{11}-2\right)+22+6\sqrt{11} by \sqrt{20+6\sqrt{11}}.

$$x=\frac{\left(\sqrt{11}-3\right)\left(\left(\sqrt{11}-2\right)y+6\sqrt{11}+22\right)}{2}$$

Solve for y

$$y=\frac{5\sqrt{11}x+17x-34\sqrt{11}-110}{7}$$

Steps for Solving Linear Equation

Factor 176=4^{2}\times 11. Rewrite the square root of the product \sqrt{4^{2}\times 11} as the product of square roots \sqrt{4^{2}}\sqrt{11}. Take the square root of 4^{2}.

$$y\sqrt{15-4\sqrt{11}}=\left(x-\sqrt{44}\right)\sqrt{20+\sqrt{396}}$$

Factor 44=2^{2}\times 11. Rewrite the square root of the product \sqrt{2^{2}\times 11} as the product of square roots \sqrt{2^{2}}\sqrt{11}. Take the square root of 2^{2}.

$$y\sqrt{15-4\sqrt{11}}=\left(x-2\sqrt{11}\right)\sqrt{20+\sqrt{396}}$$

Factor 396=6^{2}\times 11. Rewrite the square root of the product \sqrt{6^{2}\times 11} as the product of square roots \sqrt{6^{2}}\sqrt{11}. Take the square root of 6^{2}.

$$y\sqrt{15-4\sqrt{11}}=\left(x-2\sqrt{11}\right)\sqrt{20+6\sqrt{11}}$$

Use the distributive property to multiply x-2\sqrt{11} by \sqrt{20+6\sqrt{11}}.

$$y\sqrt{15-4\sqrt{11}}=x\sqrt{20+6\sqrt{11}}-2\sqrt{11}\sqrt{20+6\sqrt{11}}$$

The equation is in standard form.

$$\sqrt{15-4\sqrt{11}}y=\sqrt{6\sqrt{11}+20}x-2\sqrt{11}\sqrt{6\sqrt{11}+20}$$

Divide both sides by \sqrt{15-4\sqrt{11}}.

$$\frac{\sqrt{15-4\sqrt{11}}y}{\sqrt{15-4\sqrt{11}}}=\frac{\left(\sqrt{11}+3\right)\left(x-2\sqrt{11}\right)}{\sqrt{15-4\sqrt{11}}}$$

Dividing by \sqrt{15-4\sqrt{11}} undoes the multiplication by \sqrt{15-4\sqrt{11}}.

$$y=\frac{\left(\sqrt{11}+3\right)\left(x-2\sqrt{11}\right)}{\sqrt{15-4\sqrt{11}}}$$

Divide \left(x-2\sqrt{11}\right)\left(\sqrt{11}+3\right) by \sqrt{15-4\sqrt{11}}.

$$y=\frac{5\sqrt{11}+17}{7}\left(x-2\sqrt{11}\right)$$