Solve a+66/36-a^2-a+11/a+6+a-11/6-a

$$\frac { a + 66 } { 36 - a ^ { 2 } } - \frac { a + 11 } { a + 6 } + \frac { a - 11 } { 6 - a }$$

Evaluate

$$\frac{2a-11}{6-a}$$

Short Solution Steps

Factor 36-a^{2}.

$$\frac{a+66}{\left(a-6\right)\left(-a-6\right)}-\frac{a+11}{a+6}+\frac{a-11}{6-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-6\right)\left(-a-6\right) and a+6 is \left(a-6\right)\left(a+6\right). Multiply \frac{a+66}{\left(a-6\right)\left(-a-6\right)} times \frac{-1}{-1}. Multiply \frac{a+11}{a+6} times \frac{a-6}{a-6}.

$$\frac{-\left(a+66\right)}{\left(a-6\right)\left(a+6\right)}-\frac{\left(a+11\right)\left(a-6\right)}{\left(a-6\right)\left(a+6\right)}+\frac{a-11}{6-a}$$

Since \frac{-\left(a+66\right)}{\left(a-6\right)\left(a+6\right)} and \frac{\left(a+11\right)\left(a-6\right)}{\left(a-6\right)\left(a+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(a+66\right)-\left(a+11\right)\left(a-6\right)}{\left(a-6\right)\left(a+6\right)}+\frac{a-11}{6-a}$$

Do the multiplications in -\left(a+66\right)-\left(a+11\right)\left(a-6\right).

$$\frac{-a-66-a^{2}+6a-11a+66}{\left(a-6\right)\left(a+6\right)}+\frac{a-11}{6-a}$$

Combine like terms in -a-66-a^{2}+6a-11a+66.

$$\frac{-6a-a^{2}}{\left(a-6\right)\left(a+6\right)}+\frac{a-11}{6-a}$$

Factor the expressions that are not already factored in \frac{-6a-a^{2}}{\left(a-6\right)\left(a+6\right)}.

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

Extract the negative sign in -6-a.

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

Cancel out a+6 in both numerator and denominator.

$$\frac{-a}{a-6}+\frac{a-11}{6-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-6 and 6-a is a-6. Multiply \frac{a-11}{6-a} times \frac{-1}{-1}.

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

Since \frac{-a}{a-6} and \frac{-\left(a-11\right)}{a-6} have the same denominator, add them by adding their numerators.

$$\frac{-a-\left(a-11\right)}{a-6}$$

Do the multiplications in -a-\left(a-11\right).

$$\frac{-a-a+11}{a-6}$$

Combine like terms in -a-a+11.

$$\frac{-2a+11}{a-6}$$

Expand

$$\frac{2a-11}{6-a}$$

Short Solution Steps

Factor 36-a^{2}.

$$\frac{a+66}{\left(a-6\right)\left(-a-6\right)}-\frac{a+11}{a+6}+\frac{a-11}{6-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-6\right)\left(-a-6\right) and a+6 is \left(a-6\right)\left(a+6\right). Multiply \frac{a+66}{\left(a-6\right)\left(-a-6\right)} times \frac{-1}{-1}. Multiply \frac{a+11}{a+6} times \frac{a-6}{a-6}.

$$\frac{-\left(a+66\right)}{\left(a-6\right)\left(a+6\right)}-\frac{\left(a+11\right)\left(a-6\right)}{\left(a-6\right)\left(a+6\right)}+\frac{a-11}{6-a}$$

Since \frac{-\left(a+66\right)}{\left(a-6\right)\left(a+6\right)} and \frac{\left(a+11\right)\left(a-6\right)}{\left(a-6\right)\left(a+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(a+66\right)-\left(a+11\right)\left(a-6\right)}{\left(a-6\right)\left(a+6\right)}+\frac{a-11}{6-a}$$

Do the multiplications in -\left(a+66\right)-\left(a+11\right)\left(a-6\right).

$$\frac{-a-66-a^{2}+6a-11a+66}{\left(a-6\right)\left(a+6\right)}+\frac{a-11}{6-a}$$

Combine like terms in -a-66-a^{2}+6a-11a+66.

$$\frac{-6a-a^{2}}{\left(a-6\right)\left(a+6\right)}+\frac{a-11}{6-a}$$

Factor the expressions that are not already factored in \frac{-6a-a^{2}}{\left(a-6\right)\left(a+6\right)}.

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

Extract the negative sign in -6-a.

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

Cancel out a+6 in both numerator and denominator.

$$\frac{-a}{a-6}+\frac{a-11}{6-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-6 and 6-a is a-6. Multiply \frac{a-11}{6-a} times \frac{-1}{-1}.

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

Since \frac{-a}{a-6} and \frac{-\left(a-11\right)}{a-6} have the same denominator, add them by adding their numerators.

$$\frac{-a-\left(a-11\right)}{a-6}$$

Do the multiplications in -a-\left(a-11\right).

$$\frac{-a-a+11}{a-6}$$

Combine like terms in -a-a+11.

$$\frac{-2a+11}{a-6}$$