Solve a/a^2-9-6/a^2-3a

$$\frac { a } { a ^ { 2 } - 9 } - \frac { 6 } { a ^ { 2 } - 3 a }$$

Evaluate

$$\frac{a^{2}-6a-18}{a\left(a^{2}-9\right)}$$

Short Solution Steps

Factor a^{2}-9. Factor a^{2}-3a.

$$\frac{a}{\left(a-3\right)\left(a+3\right)}-\frac{6}{a\left(a-3\right)}$$

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\left(a-3\right) is a\left(a-3\right)\left(a+3\right). Multiply \frac{a}{\left(a-3\right)\left(a+3\right)} times \frac{a}{a}. Multiply \frac{6}{a\left(a-3\right)} times \frac{a+3}{a+3}.

$$\frac{aa}{a\left(a-3\right)\left(a+3\right)}-\frac{6\left(a+3\right)}{a\left(a-3\right)\left(a+3\right)}$$

Since \frac{aa}{a\left(a-3\right)\left(a+3\right)} and \frac{6\left(a+3\right)}{a\left(a-3\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{aa-6\left(a+3\right)}{a\left(a-3\right)\left(a+3\right)}$$

Do the multiplications in aa-6\left(a+3\right).

$$\frac{a^{2}-6a-18}{a\left(a-3\right)\left(a+3\right)}$$

Expand a\left(a-3\right)\left(a+3\right).

$$\frac{a^{2}-6a-18}{a^{3}-9a}$$

Differentiate w.r.t. a

$$\frac{-a^{4}+12a^{3}+45a^{2}-162}{\left(a\left(a^{2}-9\right)\right)^{2}}$$