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

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

Evaluate

$$\frac{5a+3b}{a^{2}-b^{2}}$$

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in 3a+b+2\left(a+b\right).

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

Combine like terms in 3a+b+2a+2b.

$$\frac{5a+3b}{\left(a+b\right)\left(a-b\right)}$$

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

$$\frac{5a+3b}{a^{2}-b^{2}}$$

Expand

$$\frac{5a+3b}{a^{2}-b^{2}}$$

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in 3a+b+2\left(a+b\right).

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

Combine like terms in 3a+b+2a+2b.

$$\frac{5a+3b}{\left(a+b\right)\left(a-b\right)}$$

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

$$\frac{5a+3b}{a^{2}-b^{2}}$$