Solve 2a+b/2a^2-ab-16a/4a^2-b^2+2a-b/2a^2+ab

$$\frac { 2 a + b } { 2 a ^ { 2 } - a b } - \frac { 16 a } { 4 a ^ { 2 } - b ^ { 2 } } + \frac { 2 a - b } { 2 a ^ { 2 } + a b }$$

Evaluate

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

Short Solution Steps

Factor 2a^{2}-ab. Factor 4a^{2}-b^{2}.

$$\frac{2a+b}{a\left(2a-b\right)}-\frac{16a}{\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(2a-b\right) and \left(2a+b\right)\left(2a-b\right) is a\left(2a+b\right)\left(2a-b\right). Multiply \frac{2a+b}{a\left(2a-b\right)} times \frac{2a+b}{2a+b}. Multiply \frac{16a}{\left(2a+b\right)\left(2a-b\right)} times \frac{a}{a}.

$$\frac{\left(2a+b\right)\left(2a+b\right)}{a\left(2a+b\right)\left(2a-b\right)}-\frac{16aa}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Since \frac{\left(2a+b\right)\left(2a+b\right)}{a\left(2a+b\right)\left(2a-b\right)} and \frac{16aa}{a\left(2a+b\right)\left(2a-b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2a+b\right)\left(2a+b\right)-16aa}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Do the multiplications in \left(2a+b\right)\left(2a+b\right)-16aa.

$$\frac{4a^{2}+2ab+2ab+b^{2}-16a^{2}}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Combine like terms in 4a^{2}+2ab+2ab+b^{2}-16a^{2}.

$$\frac{-12a^{2}+b^{2}+4ab}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Factor the expressions that are not already factored in \frac{-12a^{2}+b^{2}+4ab}{a\left(2a+b\right)\left(2a-b\right)}.

$$\frac{\left(-6a-b\right)\left(2a-b\right)}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Cancel out 2a-b in both numerator and denominator.

$$\frac{-6a-b}{a\left(2a+b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Factor 2a^{2}+ab.

$$\frac{-6a-b}{a\left(2a+b\right)}+\frac{2a-b}{a\left(2a+b\right)}$$

Since \frac{-6a-b}{a\left(2a+b\right)} and \frac{2a-b}{a\left(2a+b\right)} have the same denominator, add them by adding their numerators.

$$\frac{-6a-b+2a-b}{a\left(2a+b\right)}$$

Combine like terms in -6a-b+2a-b.

$$\frac{-4a-2b}{a\left(2a+b\right)}$$

Factor the expressions that are not already factored in \frac{-4a-2b}{a\left(2a+b\right)}.

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

Extract the negative sign in -2a-b.

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

Cancel out 2a+b in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor 2a^{2}-ab. Factor 4a^{2}-b^{2}.

$$\frac{2a+b}{a\left(2a-b\right)}-\frac{16a}{\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(2a-b\right) and \left(2a+b\right)\left(2a-b\right) is a\left(2a+b\right)\left(2a-b\right). Multiply \frac{2a+b}{a\left(2a-b\right)} times \frac{2a+b}{2a+b}. Multiply \frac{16a}{\left(2a+b\right)\left(2a-b\right)} times \frac{a}{a}.

$$\frac{\left(2a+b\right)\left(2a+b\right)}{a\left(2a+b\right)\left(2a-b\right)}-\frac{16aa}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Since \frac{\left(2a+b\right)\left(2a+b\right)}{a\left(2a+b\right)\left(2a-b\right)} and \frac{16aa}{a\left(2a+b\right)\left(2a-b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2a+b\right)\left(2a+b\right)-16aa}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Do the multiplications in \left(2a+b\right)\left(2a+b\right)-16aa.

$$\frac{4a^{2}+2ab+2ab+b^{2}-16a^{2}}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Combine like terms in 4a^{2}+2ab+2ab+b^{2}-16a^{2}.

$$\frac{-12a^{2}+b^{2}+4ab}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Factor the expressions that are not already factored in \frac{-12a^{2}+b^{2}+4ab}{a\left(2a+b\right)\left(2a-b\right)}.

$$\frac{\left(-6a-b\right)\left(2a-b\right)}{a\left(2a+b\right)\left(2a-b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Cancel out 2a-b in both numerator and denominator.

$$\frac{-6a-b}{a\left(2a+b\right)}+\frac{2a-b}{2a^{2}+ab}$$

Factor 2a^{2}+ab.

$$\frac{-6a-b}{a\left(2a+b\right)}+\frac{2a-b}{a\left(2a+b\right)}$$

Since \frac{-6a-b}{a\left(2a+b\right)} and \frac{2a-b}{a\left(2a+b\right)} have the same denominator, add them by adding their numerators.

$$\frac{-6a-b+2a-b}{a\left(2a+b\right)}$$

Combine like terms in -6a-b+2a-b.

$$\frac{-4a-2b}{a\left(2a+b\right)}$$

Factor the expressions that are not already factored in \frac{-4a-2b}{a\left(2a+b\right)}.

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

Extract the negative sign in -2a-b.

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

Cancel out 2a+b in both numerator and denominator.

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