Solve frac{2sqrt{2}}{3-sqrt{5}}=frac{3+sqrt{5}}{x}

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

Solve for x

$$x=\sqrt{2}\approx 1.414213562$$

Steps for Solving Linear Equation

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

$$\left(\frac{1}{4}x\times 5^{\frac{1}{2}}+\frac{3}{4}x\right)\times 2\sqrt{2}=3+\sqrt{5}$$

Use the distributive property to multiply \frac{1}{4}x\times 5^{\frac{1}{2}}+\frac{3}{4}x by 2.

$$\left(\frac{1}{4}x\times 5^{\frac{1}{2}}\times 2+\frac{3}{4}x\times 2\right)\sqrt{2}=3+\sqrt{5}$$

Multiply \frac{1}{4} and 2 to get \frac{2}{4}.

$$\left(\frac{2}{4}x\times 5^{\frac{1}{2}}+\frac{3}{4}x\times 2\right)\sqrt{2}=3+\sqrt{5}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\left(\frac{1}{2}x\times 5^{\frac{1}{2}}+\frac{3}{4}x\times 2\right)\sqrt{2}=3+\sqrt{5}$$

Express \frac{3}{4}\times 2 as a single fraction.

$$\left(\frac{1}{2}x\times 5^{\frac{1}{2}}+\frac{3\times 2}{4}x\right)\sqrt{2}=3+\sqrt{5}$$

Multiply 3 and 2 to get 6.

$$\left(\frac{1}{2}x\times 5^{\frac{1}{2}}+\frac{6}{4}x\right)\sqrt{2}=3+\sqrt{5}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$\left(\frac{1}{2}x\times 5^{\frac{1}{2}}+\frac{3}{2}x\right)\sqrt{2}=3+\sqrt{5}$$

Use the distributive property to multiply \frac{1}{2}x\times 5^{\frac{1}{2}}+\frac{3}{2}x by \sqrt{2}.

$$\frac{1}{2}x\times 5^{\frac{1}{2}}\sqrt{2}+\frac{3}{2}x\sqrt{2}=3+\sqrt{5}$$

Reorder the terms.

$$\frac{1}{2}\sqrt{2}\sqrt{5}x+\frac{3}{2}\sqrt{2}x=\sqrt{5}+3$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{1}{2}\sqrt{10}x+\frac{3}{2}\sqrt{2}x=\sqrt{5}+3$$

Combine all terms containing x.

$$\left(\frac{1}{2}\sqrt{10}+\frac{3}{2}\sqrt{2}\right)x=\sqrt{5}+3$$

The equation is in standard form.

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

Divide both sides by \frac{1}{2}\sqrt{10}+\frac{3}{2}\sqrt{2}.

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

Dividing by \frac{1}{2}\sqrt{10}+\frac{3}{2}\sqrt{2} undoes the multiplication by \frac{1}{2}\sqrt{10}+\frac{3}{2}\sqrt{2}.

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

Divide 3+\sqrt{5} by \frac{1}{2}\sqrt{10}+\frac{3}{2}\sqrt{2}.

$$x=\sqrt{2}$$