Solve (1/a-2-a-2)diva+3/a-2

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

Evaluate

$$\frac{5-a^{2}}{a+3}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -a-2 times \frac{a-2}{a-2}.

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

Since \frac{1}{a-2} and \frac{\left(-a-2\right)\left(a-2\right)}{a-2} have the same denominator, add them by adding their numerators.

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

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

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

Combine like terms in 1-a^{2}+2a-2a+4.

$$\frac{\frac{5-a^{2}}{a-2}}{\frac{a+3}{a-2}}$$

Divide \frac{5-a^{2}}{a-2} by \frac{a+3}{a-2} by multiplying \frac{5-a^{2}}{a-2} by the reciprocal of \frac{a+3}{a-2}.

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

Cancel out a-2 in both numerator and denominator.

$$\frac{-a^{2}+5}{a+3}$$

Expand

$$\frac{5-a^{2}}{a+3}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -a-2 times \frac{a-2}{a-2}.

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

Since \frac{1}{a-2} and \frac{\left(-a-2\right)\left(a-2\right)}{a-2} have the same denominator, add them by adding their numerators.

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

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

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

Combine like terms in 1-a^{2}+2a-2a+4.

$$\frac{\frac{5-a^{2}}{a-2}}{\frac{a+3}{a-2}}$$

Divide \frac{5-a^{2}}{a-2} by \frac{a+3}{a-2} by multiplying \frac{5-a^{2}}{a-2} by the reciprocal of \frac{a+3}{a-2}.

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

Cancel out a-2 in both numerator and denominator.

$$\frac{-a^{2}+5}{a+3}$$