Solve 3/a-4+2/a+3-21/a^2-a-12

$$\frac { 3 } { a - 4 } + \frac { 2 } { a + 3 } - \frac { 21 } { a ^ { 2 } - a - 12 }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-4 and a+3 is \left(a-4\right)\left(a+3\right). Multiply \frac{3}{a-4} times \frac{a+3}{a+3}. Multiply \frac{2}{a+3} times \frac{a-4}{a-4}.

$$\frac{3\left(a+3\right)}{\left(a-4\right)\left(a+3\right)}+\frac{2\left(a-4\right)}{\left(a-4\right)\left(a+3\right)}-\frac{21}{a^{2}-a-12}$$

Since \frac{3\left(a+3\right)}{\left(a-4\right)\left(a+3\right)} and \frac{2\left(a-4\right)}{\left(a-4\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(a+3\right)+2\left(a-4\right)}{\left(a-4\right)\left(a+3\right)}-\frac{21}{a^{2}-a-12}$$

Do the multiplications in 3\left(a+3\right)+2\left(a-4\right).

$$\frac{3a+9+2a-8}{\left(a-4\right)\left(a+3\right)}-\frac{21}{a^{2}-a-12}$$

Combine like terms in 3a+9+2a-8.

$$\frac{5a+1}{\left(a-4\right)\left(a+3\right)}-\frac{21}{a^{2}-a-12}$$

Factor a^{2}-a-12.

$$\frac{5a+1}{\left(a-4\right)\left(a+3\right)}-\frac{21}{\left(a-4\right)\left(a+3\right)}$$

Since \frac{5a+1}{\left(a-4\right)\left(a+3\right)} and \frac{21}{\left(a-4\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5a+1-21}{\left(a-4\right)\left(a+3\right)}$$

Combine like terms in 5a+1-21.

$$\frac{5a-20}{\left(a-4\right)\left(a+3\right)}$$

Factor the expressions that are not already factored in \frac{5a-20}{\left(a-4\right)\left(a+3\right)}.

$$\frac{5\left(a-4\right)}{\left(a-4\right)\left(a+3\right)}$$

Cancel out a-4 in both numerator and denominator.

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

Differentiate w.r.t. a

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