Solve 1/a^2-3-1/a-3

$$\frac { 1 } { a ^ { 2 } - 3 } - \frac { 1 } { a - 3 }$$

Evaluate

$$\frac{a\left(1-a\right)}{\left(a-3\right)\left(a^{2}-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a^{2}-3 and a-3 is \left(a-3\right)\left(a^{2}-3\right). Multiply \frac{1}{a^{2}-3} times \frac{a-3}{a-3}. Multiply \frac{1}{a-3} times \frac{a^{2}-3}{a^{2}-3}.

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

Since \frac{a-3}{\left(a-3\right)\left(a^{2}-3\right)} and \frac{a^{2}-3}{\left(a-3\right)\left(a^{2}-3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a-3-\left(a^{2}-3\right).

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

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

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

Expand \left(a-3\right)\left(a^{2}-3\right).

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

Differentiate w.r.t. a

$$\frac{a^{4}-2a^{3}+6a^{2}-18a+9}{\left(\left(a-3\right)\left(a^{2}-3\right)\right)^{2}}$$