Solve frac{2x+sqrt{5}}{sqrt{3}}-frac{x-2sqrt{5}}{sqrt{12}}=frac{sqrt{5}(sqrt{27}+3)}{6}

$$\frac { 2 x + \sqrt { 5 } } { \sqrt { 3 } } - \frac { x - 2 \sqrt { 5 } } { \sqrt { 12 } } = \frac { \sqrt { 5 } ( \sqrt { 27 } + 3 ) } { 6 }$$

Solve for x

$$x=\frac{\sqrt{15}-\sqrt{5}}{3}\approx 0.545638456$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6.

$$2\times 3^{\frac{1}{2}}\left(2x+\sqrt{5}\right)-3^{\frac{1}{2}}\left(x-2\sqrt{5}\right)=\sqrt{5}\left(\sqrt{27}+3\right)$$

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

$$2\times 3^{\frac{1}{2}}\times 2x+2\times 3^{\frac{1}{2}}\sqrt{5}-3^{\frac{1}{2}}\left(x-2\sqrt{5}\right)=\sqrt{5}\left(\sqrt{27}+3\right)$$

Multiply 2 and 2 to get 4.

$$4\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}\sqrt{5}-3^{\frac{1}{2}}\left(x-2\sqrt{5}\right)=\sqrt{5}\left(\sqrt{27}+3\right)$$

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

$$4\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}\sqrt{5}-\left(3^{\frac{1}{2}}x+3^{\frac{1}{2}}\left(-2\right)\sqrt{5}\right)=\sqrt{5}\left(\sqrt{27}+3\right)$$

To find the opposite of 3^{\frac{1}{2}}x+3^{\frac{1}{2}}\left(-2\right)\sqrt{5}, find the opposite of each term.

$$4\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}\sqrt{5}-3^{\frac{1}{2}}x-3^{\frac{1}{2}}\left(-2\right)\sqrt{5}=\sqrt{5}\left(\sqrt{27}+3\right)$$

Multiply -1 and -2 to get 2.

$$4\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}\sqrt{5}-3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}\sqrt{5}=\sqrt{5}\left(\sqrt{27}+3\right)$$

Combine 4\times 3^{\frac{1}{2}}x and -3^{\frac{1}{2}}x to get 3\times 3^{\frac{1}{2}}x.

$$3\times 3^{\frac{1}{2}}x+2\times 3^{\frac{1}{2}}\sqrt{5}+2\times 3^{\frac{1}{2}}\sqrt{5}=\sqrt{5}\left(\sqrt{27}+3\right)$$

Combine 2\times 3^{\frac{1}{2}}\sqrt{5} and 2\times 3^{\frac{1}{2}}\sqrt{5} to get 4\times 3^{\frac{1}{2}}\sqrt{5}.

$$3\times 3^{\frac{1}{2}}x+4\times 3^{\frac{1}{2}}\sqrt{5}=\sqrt{5}\left(\sqrt{27}+3\right)$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

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

Use the distributive property to multiply \sqrt{5} by 3\sqrt{3}+3.

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

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$3\times 3^{\frac{1}{2}}x+4\times 3^{\frac{1}{2}}\sqrt{5}=3\sqrt{15}+3\sqrt{5}$$

To multiply powers of the same base, add their exponents. Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$3^{\frac{3}{2}}x+4\times 3^{\frac{1}{2}}\sqrt{5}=3\sqrt{15}+3\sqrt{5}$$

Subtract 4\times 3^{\frac{1}{2}}\sqrt{5} from both sides.

$$3^{\frac{3}{2}}x=3\sqrt{15}+3\sqrt{5}-4\times 3^{\frac{1}{2}}\sqrt{5}$$

Reorder the terms.

$$3^{\frac{3}{2}}x=3\sqrt{5}+3\sqrt{15}-4\sqrt{3}\sqrt{5}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$3^{\frac{3}{2}}x=3\sqrt{5}+3\sqrt{15}-4\sqrt{15}$$

Combine 3\sqrt{15} and -4\sqrt{15} to get -\sqrt{15}.

$$3^{\frac{3}{2}}x=3\sqrt{5}-\sqrt{15}$$

The equation is in standard form.

$$3\sqrt{3}x=3\sqrt{5}-\sqrt{15}$$

Divide both sides by 3\times 3^{\frac{1}{2}}.

$$\frac{3\sqrt{3}x}{3\sqrt{3}}=\frac{3\sqrt{5}-\sqrt{15}}{3\sqrt{3}}$$

Dividing by 3\times 3^{\frac{1}{2}} undoes the multiplication by 3\times 3^{\frac{1}{2}}.

$$x=\frac{3\sqrt{5}-\sqrt{15}}{3\sqrt{3}}$$

Divide 3\sqrt{5}-\sqrt{15} by 3\times 3^{\frac{1}{2}}.

$$x=\frac{\sqrt{15}-\sqrt{5}}{3}$$