Solve n/3+n=sqrt{3/8}*3

$$\frac{ n }{ 3+n } = \sqrt{ \frac{ 3 }{ 8 } } \times 3$$

Solve for n

$$n=\frac{-18\sqrt{6}-81}{19}\approx -6.583727125$$

Steps for Solving Linear Equation

Variable n cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by n+3.

$$n=3\sqrt{\frac{3}{8}}\left(n+3\right)$$

Rewrite the square root of the division \sqrt{\frac{3}{8}} as the division of square roots \frac{\sqrt{3}}{\sqrt{8}}.

$$n=3\times \frac{\sqrt{3}}{\sqrt{8}}\left(n+3\right)$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$n=3\times \frac{\sqrt{3}}{2\sqrt{2}}\left(n+3\right)$$

Rationalize the denominator of \frac{\sqrt{3}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

$$n=3\times \frac{\sqrt{3}\sqrt{2}}{2\times 2}\left(n+3\right)$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$n=3\times \frac{\sqrt{6}}{2\times 2}\left(n+3\right)$$

Multiply 2 and 2 to get 4.

$$n=3\times \frac{\sqrt{6}}{4}\left(n+3\right)$$

Express 3\times \frac{\sqrt{6}}{4} as a single fraction.

$$n=\frac{3\sqrt{6}}{4}\left(n+3\right)$$

Express \frac{3\sqrt{6}}{4}\left(n+3\right) as a single fraction.

$$n=\frac{3\sqrt{6}\left(n+3\right)}{4}$$

Use the distributive property to multiply 3\sqrt{6} by n+3.

$$n=\frac{3\sqrt{6}n+9\sqrt{6}}{4}$$

Subtract \frac{3\sqrt{6}n+9\sqrt{6}}{4} from both sides.

$$n-\frac{3\sqrt{6}n+9\sqrt{6}}{4}=0$$

Multiply both sides of the equation by 4.

$$4n-\left(3\sqrt{6}n+9\sqrt{6}\right)=0$$

To find the opposite of 3\sqrt{6}n+9\sqrt{6}, find the opposite of each term.

$$4n-3\sqrt{6}n-9\sqrt{6}=0$$

Add 9\sqrt{6} to both sides. Anything plus zero gives itself.

$$4n-3\sqrt{6}n=9\sqrt{6}$$

Combine all terms containing n.

$$\left(4-3\sqrt{6}\right)n=9\sqrt{6}$$

Divide both sides by 4-3\sqrt{6}.

$$\frac{\left(4-3\sqrt{6}\right)n}{4-3\sqrt{6}}=\frac{9\sqrt{6}}{4-3\sqrt{6}}$$

Dividing by 4-3\sqrt{6} undoes the multiplication by 4-3\sqrt{6}.

$$n=\frac{9\sqrt{6}}{4-3\sqrt{6}}$$

Divide 9\sqrt{6} by 4-3\sqrt{6}.

$$n=\frac{-18\sqrt{6}-81}{19}$$