Solve (a+2/a^2-2a-a-1/a^2-4a+4)div4-a/a^2-2a

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

Evaluate

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

Short Solution Steps

Factor a^{2}-2a. Factor a^{2}-4a+4.

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

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

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

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

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

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

$$\frac{\frac{a^{2}-2a+2a-4-a^{2}+a}{a\left(a-2\right)^{2}}}{\frac{4-a}{a^{2}-2a}}$$

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

$$\frac{\frac{a-4}{a\left(a-2\right)^{2}}}{\frac{4-a}{a^{2}-2a}}$$

Divide \frac{a-4}{a\left(a-2\right)^{2}} by \frac{4-a}{a^{2}-2a} by multiplying \frac{a-4}{a\left(a-2\right)^{2}} by the reciprocal of \frac{4-a}{a^{2}-2a}.

$$\frac{\left(a-4\right)\left(a^{2}-2a\right)}{a\left(a-2\right)^{2}\left(4-a\right)}$$

Extract the negative sign in a-4.

$$\frac{-\left(-a+4\right)\left(a^{2}-2a\right)}{a\left(-a+4\right)\left(a-2\right)^{2}}$$

Cancel out -a+4 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

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

Expand

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

Short Solution Steps

Factor a^{2}-2a. Factor a^{2}-4a+4.

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

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

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

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

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

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

$$\frac{\frac{a^{2}-2a+2a-4-a^{2}+a}{a\left(a-2\right)^{2}}}{\frac{4-a}{a^{2}-2a}}$$

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

$$\frac{\frac{a-4}{a\left(a-2\right)^{2}}}{\frac{4-a}{a^{2}-2a}}$$

Divide \frac{a-4}{a\left(a-2\right)^{2}} by \frac{4-a}{a^{2}-2a} by multiplying \frac{a-4}{a\left(a-2\right)^{2}} by the reciprocal of \frac{4-a}{a^{2}-2a}.

$$\frac{\left(a-4\right)\left(a^{2}-2a\right)}{a\left(a-2\right)^{2}\left(4-a\right)}$$

Extract the negative sign in a-4.

$$\frac{-\left(-a+4\right)\left(a^{2}-2a\right)}{a\left(-a+4\right)\left(a-2\right)^{2}}$$

Cancel out -a+4 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

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