Solve -1-3/2n-1/6n^2

$$- 1 - \frac { 3 } { 2 n } - \frac { 1 } { 6 n ^ { 2 } }$$

Evaluate

$$-1-\frac{3}{2n}-\frac{1}{6n^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -1 times \frac{2n}{2n}.

$$-\frac{2n}{2n}-\frac{3}{2n}-\frac{1}{6n^{2}}$$

Since -\frac{2n}{2n} and \frac{3}{2n} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2n-3}{2n}-\frac{1}{6n^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2n and 6n^{2} is 6n^{2}. Multiply \frac{-2n-3}{2n} times \frac{3n}{3n}.

$$\frac{\left(-2n-3\right)\times 3n}{6n^{2}}-\frac{1}{6n^{2}}$$

Since \frac{\left(-2n-3\right)\times 3n}{6n^{2}} and \frac{1}{6n^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-2n-3\right)\times 3n-1}{6n^{2}}$$

Do the multiplications in \left(-2n-3\right)\times 3n-1.

$$\frac{-6n^{2}-9n-1}{6n^{2}}$$

Factor the expressions that are not already factored in \frac{-6n^{2}-9n-1}{6n^{2}}.

$$\frac{-6\left(n-\left(-\frac{1}{12}\sqrt{57}-\frac{3}{4}\right)\right)\left(n-\left(\frac{1}{12}\sqrt{57}-\frac{3}{4}\right)\right)}{6n^{2}}$$

Cancel out 6 in both numerator and denominator.

$$\frac{-\left(n-\left(-\frac{1}{12}\sqrt{57}-\frac{3}{4}\right)\right)\left(n-\left(\frac{1}{12}\sqrt{57}-\frac{3}{4}\right)\right)}{n^{2}}$$

To find the opposite of -\frac{1}{12}\sqrt{57}-\frac{3}{4}, find the opposite of each term.

$$\frac{-\left(n+\frac{1}{12}\sqrt{57}+\frac{3}{4}\right)\left(n-\left(\frac{1}{12}\sqrt{57}-\frac{3}{4}\right)\right)}{n^{2}}$$

To find the opposite of \frac{1}{12}\sqrt{57}-\frac{3}{4}, find the opposite of each term.

$$\frac{-\left(n+\frac{1}{12}\sqrt{57}+\frac{3}{4}\right)\left(n-\frac{1}{12}\sqrt{57}+\frac{3}{4}\right)}{n^{2}}$$

Use the distributive property to multiply -1 by n+\frac{1}{12}\sqrt{57}+\frac{3}{4}.

$$\frac{\left(-n-\frac{1}{12}\sqrt{57}-\frac{3}{4}\right)\left(n-\frac{1}{12}\sqrt{57}+\frac{3}{4}\right)}{n^{2}}$$

Use the distributive property to multiply -n-\frac{1}{12}\sqrt{57}-\frac{3}{4} by n-\frac{1}{12}\sqrt{57}+\frac{3}{4} and combine like terms.

$$\frac{-n^{2}-\frac{3}{2}n+\frac{1}{144}\left(\sqrt{57}\right)^{2}-\frac{9}{16}}{n^{2}}$$

The square of \sqrt{57} is 57.

$$\frac{-n^{2}-\frac{3}{2}n+\frac{1}{144}\times 57-\frac{9}{16}}{n^{2}}$$

Multiply \frac{1}{144} and 57 to get \frac{19}{48}.

$$\frac{-n^{2}-\frac{3}{2}n+\frac{19}{48}-\frac{9}{16}}{n^{2}}$$

Subtract \frac{9}{16} from \frac{19}{48} to get -\frac{1}{6}.

$$\frac{-n^{2}-\frac{3}{2}n-\frac{1}{6}}{n^{2}}$$

Factor

$$-\frac{\left(n-\left(-\frac{\sqrt{57}}{12}-\frac{3}{4}\right)\right)\left(n-\left(\frac{\sqrt{57}}{12}-\frac{3}{4}\right)\right)}{n^{2}}$$