Solve 2a/a+1+(3/(a-1)^2-3/a^2-1):3/a^2-2a+1

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

Evaluate

$$2$$

Short Solution Steps

Factor a^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)^{2} and \left(a-1\right)\left(a+1\right) is \left(a+1\right)\left(a-1\right)^{2}. Multiply \frac{3}{\left(a-1\right)^{2}} times \frac{a+1}{a+1}. Multiply \frac{3}{\left(a-1\right)\left(a+1\right)} times \frac{a-1}{a-1}.

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

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

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

Do the multiplications in 3\left(a+1\right)-3\left(a-1\right).

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

Combine like terms in 3a+3-3a+3.

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

Divide \frac{6}{\left(a+1\right)\left(a-1\right)^{2}} by \frac{3}{a^{2}-2a+1} by multiplying \frac{6}{\left(a+1\right)\left(a-1\right)^{2}} by the reciprocal of \frac{3}{a^{2}-2a+1}.

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

Cancel out 3 in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{2\left(a^{2}-2a+1\right)}{\left(a+1\right)\left(a-1\right)^{2}}.

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

Cancel out \left(a-1\right)^{2} in both numerator and denominator.

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

Since \frac{2a}{a+1} and \frac{2}{a+1} have the same denominator, add them by adding their numerators.

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

Factor the expressions that are not already factored in \frac{2a+2}{a+1}.

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

Cancel out a+1 in both numerator and denominator.

$$2$$

Factor

$$2$$