Solve n^2-25/n^2-36-n+5/n-6

$$\frac { n ^ { 2 } - 25 } { n ^ { 2 } - 36 } - \frac { n + 5 } { n - 6 }$$

Evaluate

$$-\frac{11\left(n+5\right)}{n^{2}-36}$$

Short Solution Steps

Factor n^{2}-36.

$$\frac{n^{2}-25}{\left(n-6\right)\left(n+6\right)}-\frac{n+5}{n-6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(n-6\right)\left(n+6\right) and n-6 is \left(n-6\right)\left(n+6\right). Multiply \frac{n+5}{n-6} times \frac{n+6}{n+6}.

$$\frac{n^{2}-25}{\left(n-6\right)\left(n+6\right)}-\frac{\left(n+5\right)\left(n+6\right)}{\left(n-6\right)\left(n+6\right)}$$

Since \frac{n^{2}-25}{\left(n-6\right)\left(n+6\right)} and \frac{\left(n+5\right)\left(n+6\right)}{\left(n-6\right)\left(n+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{n^{2}-25-\left(n+5\right)\left(n+6\right)}{\left(n-6\right)\left(n+6\right)}$$

Do the multiplications in n^{2}-25-\left(n+5\right)\left(n+6\right).

$$\frac{n^{2}-25-n^{2}-6n-5n-30}{\left(n-6\right)\left(n+6\right)}$$

Combine like terms in n^{2}-25-n^{2}-6n-5n-30.

$$\frac{-55-11n}{\left(n-6\right)\left(n+6\right)}$$

Expand \left(n-6\right)\left(n+6\right).

$$\frac{-55-11n}{n^{2}-36}$$

Expand

$$-\frac{11\left(n+5\right)}{n^{2}-36}$$

Short Solution Steps

Factor n^{2}-36.

$$\frac{n^{2}-25}{\left(n-6\right)\left(n+6\right)}-\frac{n+5}{n-6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(n-6\right)\left(n+6\right) and n-6 is \left(n-6\right)\left(n+6\right). Multiply \frac{n+5}{n-6} times \frac{n+6}{n+6}.

$$\frac{n^{2}-25}{\left(n-6\right)\left(n+6\right)}-\frac{\left(n+5\right)\left(n+6\right)}{\left(n-6\right)\left(n+6\right)}$$

Since \frac{n^{2}-25}{\left(n-6\right)\left(n+6\right)} and \frac{\left(n+5\right)\left(n+6\right)}{\left(n-6\right)\left(n+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{n^{2}-25-\left(n+5\right)\left(n+6\right)}{\left(n-6\right)\left(n+6\right)}$$

Do the multiplications in n^{2}-25-\left(n+5\right)\left(n+6\right).

$$\frac{n^{2}-25-n^{2}-6n-5n-30}{\left(n-6\right)\left(n+6\right)}$$

Combine like terms in n^{2}-25-n^{2}-6n-5n-30.

$$\frac{-55-11n}{\left(n-6\right)\left(n+6\right)}$$

Expand \left(n-6\right)\left(n+6\right).

$$\frac{-55-11n}{n^{2}-36}$$