Solve 3a/a+b+ab-5a^2/a^2-b^2+2a/a-b

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

Evaluate

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Short Solution Steps

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

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

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

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

Since \frac{3a\left(a-b\right)}{\left(a+b\right)\left(a-b\right)} and \frac{ab-5a^{2}}{\left(a+b\right)\left(a-b\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3a\left(a-b\right)+ab-5a^{2}.

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

Combine like terms in 3a^{2}-3ab+ab-5a^{2}.

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

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

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

Extract the negative sign in -a-b.

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

Cancel out a+b in both numerator and denominator.

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

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

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

Combine like terms in -2a+2a.

$$\frac{0}{a-b}$$

Zero divided by any non-zero term gives zero.

$$0$$

Factor

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