Solve a+15/9-a^2-a+5/a+3+a-5/3-a

$$\frac { a + 15 } { 9 - a ^ { 2 } } - \frac { a + 5 } { a + 3 } + \frac { a - 5 } { 3 - a }$$

Evaluate

$$\frac{2a-5}{3-a}$$

Short Solution Steps

Factor 9-a^{2}.

$$\frac{a+15}{\left(a-3\right)\left(-a-3\right)}-\frac{a+5}{a+3}+\frac{a-5}{3-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-3\right)\left(-a-3\right) and a+3 is \left(a-3\right)\left(a+3\right). Multiply \frac{a+15}{\left(a-3\right)\left(-a-3\right)} times \frac{-1}{-1}. Multiply \frac{a+5}{a+3} times \frac{a-3}{a-3}.

$$\frac{-\left(a+15\right)}{\left(a-3\right)\left(a+3\right)}-\frac{\left(a+5\right)\left(a-3\right)}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Since \frac{-\left(a+15\right)}{\left(a-3\right)\left(a+3\right)} and \frac{\left(a+5\right)\left(a-3\right)}{\left(a-3\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(a+15\right)-\left(a+5\right)\left(a-3\right)}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Do the multiplications in -\left(a+15\right)-\left(a+5\right)\left(a-3\right).

$$\frac{-a-15-a^{2}+3a-5a+15}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Combine like terms in -a-15-a^{2}+3a-5a+15.

$$\frac{-3a-a^{2}}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Factor the expressions that are not already factored in \frac{-3a-a^{2}}{\left(a-3\right)\left(a+3\right)}.

$$\frac{a\left(-a-3\right)}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Extract the negative sign in -3-a.

$$\frac{-a\left(a+3\right)}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Cancel out a+3 in both numerator and denominator.

$$\frac{-a}{a-3}+\frac{a-5}{3-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-3 and 3-a is a-3. Multiply \frac{a-5}{3-a} times \frac{-1}{-1}.

$$\frac{-a}{a-3}+\frac{-\left(a-5\right)}{a-3}$$

Since \frac{-a}{a-3} and \frac{-\left(a-5\right)}{a-3} have the same denominator, add them by adding their numerators.

$$\frac{-a-\left(a-5\right)}{a-3}$$

Do the multiplications in -a-\left(a-5\right).

$$\frac{-a-a+5}{a-3}$$

Combine like terms in -a-a+5.

$$\frac{-2a+5}{a-3}$$

Expand

$$\frac{2a-5}{3-a}$$

Short Solution Steps

Factor 9-a^{2}.

$$\frac{a+15}{\left(a-3\right)\left(-a-3\right)}-\frac{a+5}{a+3}+\frac{a-5}{3-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-3\right)\left(-a-3\right) and a+3 is \left(a-3\right)\left(a+3\right). Multiply \frac{a+15}{\left(a-3\right)\left(-a-3\right)} times \frac{-1}{-1}. Multiply \frac{a+5}{a+3} times \frac{a-3}{a-3}.

$$\frac{-\left(a+15\right)}{\left(a-3\right)\left(a+3\right)}-\frac{\left(a+5\right)\left(a-3\right)}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Since \frac{-\left(a+15\right)}{\left(a-3\right)\left(a+3\right)} and \frac{\left(a+5\right)\left(a-3\right)}{\left(a-3\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(a+15\right)-\left(a+5\right)\left(a-3\right)}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Do the multiplications in -\left(a+15\right)-\left(a+5\right)\left(a-3\right).

$$\frac{-a-15-a^{2}+3a-5a+15}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Combine like terms in -a-15-a^{2}+3a-5a+15.

$$\frac{-3a-a^{2}}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Factor the expressions that are not already factored in \frac{-3a-a^{2}}{\left(a-3\right)\left(a+3\right)}.

$$\frac{a\left(-a-3\right)}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Extract the negative sign in -3-a.

$$\frac{-a\left(a+3\right)}{\left(a-3\right)\left(a+3\right)}+\frac{a-5}{3-a}$$

Cancel out a+3 in both numerator and denominator.

$$\frac{-a}{a-3}+\frac{a-5}{3-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-3 and 3-a is a-3. Multiply \frac{a-5}{3-a} times \frac{-1}{-1}.

$$\frac{-a}{a-3}+\frac{-\left(a-5\right)}{a-3}$$

Since \frac{-a}{a-3} and \frac{-\left(a-5\right)}{a-3} have the same denominator, add them by adding their numerators.

$$\frac{-a-\left(a-5\right)}{a-3}$$

Do the multiplications in -a-\left(a-5\right).

$$\frac{-a-a+5}{a-3}$$

Combine like terms in -a-a+5.

$$\frac{-2a+5}{a-3}$$