Solve x-a/x+a-x+a/x-a+3a^2+a/x^2-a^2=

$$\frac { x - a } { x + a } - \frac { x + a } { x - a } + \frac { 3 a ^ { 2 } + a } { x ^ { 2 } - a ^ { 2 } } =$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Since \frac{-4xa}{\left(x+a\right)\left(x-a\right)} and \frac{3a^{2}+a}{\left(x+a\right)\left(x-a\right)} have the same denominator, add them by adding their numerators.

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

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

$$\frac{-4xa+3a^{2}+a}{x^{2}-a^{2}}$$

Expand

$$\frac{a+3a^{2}-4ax}{x^{2}-a^{2}}$$

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Since \frac{-4xa}{\left(x+a\right)\left(x-a\right)} and \frac{3a^{2}+a}{\left(x+a\right)\left(x-a\right)} have the same denominator, add them by adding their numerators.

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

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

$$\frac{-4xa+3a^{2}+a}{x^{2}-a^{2}}$$