Solve a/a+2b+b/a+b

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Expand \left(a+b\right)\left(a+2b\right).

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

Differentiate w.r.t. a

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