Solve a+2/a^2+a+a-2/a-a^2-3/a^2-1

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Cancel out a in both numerator and denominator.

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

Factor a^{2}-1.

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

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

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

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

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

Do the multiplications in -2\left(-1\right)-3.

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

Do the calculations in 2-3.

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

Expand \left(a-1\right)\left(a+1\right).

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Cancel out a in both numerator and denominator.

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

Factor a^{2}-1.

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

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

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

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

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

Do the multiplications in -2\left(-1\right)-3.

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

Do the calculations in 2-3.

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

Expand \left(a-1\right)\left(a+1\right).

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