Solve a+2/a^2-2a+1-a/a^2-4a+4diva-4/a

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

Evaluate

$$\frac{16-4a-3a^{2}}{a\left(a-4\right)\left(a-2\right)^{2}}$$

Short Solution Steps

Divide \frac{1-a}{a^{2}-4a+4} by \frac{a-4}{a} by multiplying \frac{1-a}{a^{2}-4a+4} by the reciprocal of \frac{a-4}{a}.

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

Factor a^{2}-2a. Factor \left(a^{2}-4a+4\right)\left(a-4\right).

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

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-4\right)\left(a-2\right)^{2} is a\left(a-4\right)\left(a-2\right)^{2}. Multiply \frac{a+2}{a\left(a-2\right)} times \frac{\left(a-4\right)\left(a-2\right)}{\left(a-4\right)\left(a-2\right)}. Multiply \frac{\left(1-a\right)a}{\left(a-4\right)\left(a-2\right)^{2}} times \frac{a}{a}.

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

Since \frac{\left(a+2\right)\left(a-4\right)\left(a-2\right)}{a\left(a-4\right)\left(a-2\right)^{2}} and \frac{\left(1-a\right)aa}{a\left(a-4\right)\left(a-2\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(a+2\right)\left(a-4\right)\left(a-2\right)+\left(1-a\right)aa.

$$\frac{a^{3}-6a^{2}+8a+2a^{2}-12a+16+a^{2}-a^{3}}{a\left(a-4\right)\left(a-2\right)^{2}}$$

Combine like terms in a^{3}-6a^{2}+8a+2a^{2}-12a+16+a^{2}-a^{3}.

$$\frac{-3a^{2}-4a+16}{a\left(a-4\right)\left(a-2\right)^{2}}$$

Expand a\left(a-4\right)\left(a-2\right)^{2}.

$$\frac{-3a^{2}-4a+16}{a^{4}-8a^{3}+20a^{2}-16a}$$

Expand

$$\frac{16-4a-3a^{2}}{a\left(a-4\right)\left(a-2\right)^{2}}$$

Short Solution Steps

Divide \frac{1-a}{a^{2}-4a+4} by \frac{a-4}{a} by multiplying \frac{1-a}{a^{2}-4a+4} by the reciprocal of \frac{a-4}{a}.

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

Factor a^{2}-2a. Factor \left(a^{2}-4a+4\right)\left(a-4\right).

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

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-4\right)\left(a-2\right)^{2} is a\left(a-4\right)\left(a-2\right)^{2}. Multiply \frac{a+2}{a\left(a-2\right)} times \frac{\left(a-4\right)\left(a-2\right)}{\left(a-4\right)\left(a-2\right)}. Multiply \frac{\left(1-a\right)a}{\left(a-4\right)\left(a-2\right)^{2}} times \frac{a}{a}.

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

Since \frac{\left(a+2\right)\left(a-4\right)\left(a-2\right)}{a\left(a-4\right)\left(a-2\right)^{2}} and \frac{\left(1-a\right)aa}{a\left(a-4\right)\left(a-2\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(a+2\right)\left(a-4\right)\left(a-2\right)+\left(1-a\right)aa.

$$\frac{a^{3}-6a^{2}+8a+2a^{2}-12a+16+a^{2}-a^{3}}{a\left(a-4\right)\left(a-2\right)^{2}}$$

Combine like terms in a^{3}-6a^{2}+8a+2a^{2}-12a+16+a^{2}-a^{3}.

$$\frac{-3a^{2}-4a+16}{a\left(a-4\right)\left(a-2\right)^{2}}$$

Expand a\left(a-4\right)\left(a-2\right)^{2}.

$$\frac{-3a^{2}-4a+16}{a^{4}-8a^{3}+20a^{2}-16a}$$