Solve 1/5+5a+1/5-5a-1/10+10a^2

$$\frac{ 1 }{ 5+5a } + \frac{ 1 }{ 5-5a } - \frac{ 1 }{ 10+10 { a }^{ 2 } }$$

Evaluate

$$\frac{5a^{2}+3}{10\left(1-a^{4}\right)}$$

Short Solution Steps

Factor 5+5a. Factor 5-5a.

$$\frac{1}{5\left(a+1\right)}+\frac{1}{5\left(-a+1\right)}-\frac{1}{10+10a^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5\left(a+1\right) and 5\left(-a+1\right) is 5\left(a+1\right)\left(-a+1\right). Multiply \frac{1}{5\left(a+1\right)} times \frac{-a+1}{-a+1}. Multiply \frac{1}{5\left(-a+1\right)} times \frac{a+1}{a+1}.

$$\frac{-a+1}{5\left(a+1\right)\left(-a+1\right)}+\frac{a+1}{5\left(a+1\right)\left(-a+1\right)}-\frac{1}{10+10a^{2}}$$

Since \frac{-a+1}{5\left(a+1\right)\left(-a+1\right)} and \frac{a+1}{5\left(a+1\right)\left(-a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{-a+1+a+1}{5\left(a+1\right)\left(-a+1\right)}-\frac{1}{10+10a^{2}}$$

Combine like terms in -a+1+a+1.

$$\frac{2}{5\left(a+1\right)\left(-a+1\right)}-\frac{1}{10+10a^{2}}$$

Factor 10+10a^{2}.

$$\frac{2}{5\left(a+1\right)\left(-a+1\right)}-\frac{1}{10\left(a^{2}+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5\left(a+1\right)\left(-a+1\right) and 10\left(a^{2}+1\right) is 10\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right). Multiply \frac{2}{5\left(a+1\right)\left(-a+1\right)} times \frac{-2\left(a^{2}+1\right)}{-2\left(a^{2}+1\right)}. Multiply \frac{1}{10\left(a^{2}+1\right)} times \frac{\left(a-1\right)\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}.

$$\frac{2\left(-2\right)\left(a^{2}+1\right)}{10\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)}-\frac{\left(a-1\right)\left(a+1\right)}{10\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)}$$

Since \frac{2\left(-2\right)\left(a^{2}+1\right)}{10\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)} and \frac{\left(a-1\right)\left(a+1\right)}{10\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(-2\right)\left(a^{2}+1\right)-\left(a-1\right)\left(a+1\right)}{10\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)}$$

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

$$\frac{-4a^{2}-4-a^{2}-a+a+1}{10\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)}$$

Combine like terms in -4a^{2}-4-a^{2}-a+a+1.

$$\frac{-5a^{2}-3}{10\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right)}$$

Expand 10\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right).

$$\frac{-5a^{2}-3}{10a^{4}-10}$$

Factor

$$\frac{5a^{2}+3}{10\left(1-a^{4}\right)}$$