Solve a^2-b^2/a^2-a-6+1/a-3

$$\frac { a ^ { 2 } - b ^ { 2 } } { a ^ { 2 } - a - 6 } + \frac { 1 } { a - 3 }$$

Evaluate

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

Short Solution Steps

Factor a^{2}-a-6.

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

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

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

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

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

Expand \left(a-3\right)\left(a+2\right).

$$\frac{a^{2}-b^{2}+a+2}{a^{2}-a-6}$$

Expand

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

Short Solution Steps

Factor a^{2}-a-6.

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

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

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

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

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

Expand \left(a-3\right)\left(a+2\right).

$$\frac{a^{2}-b^{2}+a+2}{a^{2}-a-6}$$