Solve (a-2/a^2-4-a-1/a^2-2a)div1/a-2

$$( \frac { a - 2 } { a ^ { 2 } - 4 } - \frac { a - 1 } { a ^ { 2 } - 2 a } ) \div \frac { 1 } { a - 2 }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{a-2}{a^{2}-4}.

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

Cancel out a-2 in both numerator and denominator.

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

Factor a^{2}-2a.

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

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

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

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

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

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

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

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

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

Divide \frac{-3a+2}{a\left(a-2\right)\left(a+2\right)} by \frac{1}{a-2} by multiplying \frac{-3a+2}{a\left(a-2\right)\left(a+2\right)} by the reciprocal of \frac{1}{a-2}.

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

Cancel out a-2 in both numerator and denominator.

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

Use the distributive property to multiply a by a+2.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{a-2}{a^{2}-4}.

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

Cancel out a-2 in both numerator and denominator.

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

Factor a^{2}-2a.

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

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

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

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

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

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

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

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

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

Divide \frac{-3a+2}{a\left(a-2\right)\left(a+2\right)} by \frac{1}{a-2} by multiplying \frac{-3a+2}{a\left(a-2\right)\left(a+2\right)} by the reciprocal of \frac{1}{a-2}.

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

Cancel out a-2 in both numerator and denominator.

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

Use the distributive property to multiply a by a+2.

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