Solve a+1/a-3-a-3/a+2diva^2-6a+9/a^2-4

$$\frac { a + 1 } { a - 3 } - \frac { a - 3 } { a + 2 } \div \frac { a ^ { 2 } - 6 a + 9 } { a ^ { 2 } - 4 }$$

Evaluate

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

Short Solution Steps

Divide \frac{a-3}{a+2} by \frac{a^{2}-6a+9}{a^{2}-4} by multiplying \frac{a-3}{a+2} by the reciprocal of \frac{a^{2}-6a+9}{a^{2}-4}.

$$\frac{a+1}{a-3}-\frac{\left(a-3\right)\left(a^{2}-4\right)}{\left(a+2\right)\left(a^{2}-6a+9\right)}$$

Factor the expressions that are not already factored in \frac{\left(a-3\right)\left(a^{2}-4\right)}{\left(a+2\right)\left(a^{2}-6a+9\right)}.

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

Cancel out \left(a-3\right)\left(a+2\right) in both numerator and denominator.

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

Since \frac{a+1}{a-3} and \frac{a-2}{a-3} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a+1-\left(a-2\right).

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

Combine like terms in a+1-a+2.

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

Expand

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

Short Solution Steps

Divide \frac{a-3}{a+2} by \frac{a^{2}-6a+9}{a^{2}-4} by multiplying \frac{a-3}{a+2} by the reciprocal of \frac{a^{2}-6a+9}{a^{2}-4}.

$$\frac{a+1}{a-3}-\frac{\left(a-3\right)\left(a^{2}-4\right)}{\left(a+2\right)\left(a^{2}-6a+9\right)}$$

Factor the expressions that are not already factored in \frac{\left(a-3\right)\left(a^{2}-4\right)}{\left(a+2\right)\left(a^{2}-6a+9\right)}.

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

Cancel out \left(a-3\right)\left(a+2\right) in both numerator and denominator.

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

Since \frac{a+1}{a-3} and \frac{a-2}{a-3} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a+1-\left(a-2\right).

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

Combine like terms in a+1-a+2.

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