Solve 30a/9a^2-1+4/3a-1-5/3a+1=

$$\frac { 30 a } { 9 a ^ { 2 } - 1 } + \frac { 4 } { 3 a - 1 } - \frac { 5 } { 3 a + 1 } =$$

Evaluate

$$\frac{9}{3a-1}$$

Short Solution Steps

Factor 9a^{2}-1.

$$\frac{30a}{\left(3a-1\right)\left(3a+1\right)}+\frac{4}{3a-1}-\frac{5}{3a+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(3a-1\right)\left(3a+1\right) and 3a-1 is \left(3a-1\right)\left(3a+1\right). Multiply \frac{4}{3a-1} times \frac{3a+1}{3a+1}.

$$\frac{30a}{\left(3a-1\right)\left(3a+1\right)}+\frac{4\left(3a+1\right)}{\left(3a-1\right)\left(3a+1\right)}-\frac{5}{3a+1}$$

Since \frac{30a}{\left(3a-1\right)\left(3a+1\right)} and \frac{4\left(3a+1\right)}{\left(3a-1\right)\left(3a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{30a+4\left(3a+1\right)}{\left(3a-1\right)\left(3a+1\right)}-\frac{5}{3a+1}$$

Do the multiplications in 30a+4\left(3a+1\right).

$$\frac{30a+12a+4}{\left(3a-1\right)\left(3a+1\right)}-\frac{5}{3a+1}$$

Combine like terms in 30a+12a+4.

$$\frac{42a+4}{\left(3a-1\right)\left(3a+1\right)}-\frac{5}{3a+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(3a-1\right)\left(3a+1\right) and 3a+1 is \left(3a-1\right)\left(3a+1\right). Multiply \frac{5}{3a+1} times \frac{3a-1}{3a-1}.

$$\frac{42a+4}{\left(3a-1\right)\left(3a+1\right)}-\frac{5\left(3a-1\right)}{\left(3a-1\right)\left(3a+1\right)}$$

Since \frac{42a+4}{\left(3a-1\right)\left(3a+1\right)} and \frac{5\left(3a-1\right)}{\left(3a-1\right)\left(3a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{42a+4-5\left(3a-1\right)}{\left(3a-1\right)\left(3a+1\right)}$$

Do the multiplications in 42a+4-5\left(3a-1\right).

$$\frac{42a+4-15a+5}{\left(3a-1\right)\left(3a+1\right)}$$

Combine like terms in 42a+4-15a+5.

$$\frac{27a+9}{\left(3a-1\right)\left(3a+1\right)}$$

Factor the expressions that are not already factored in \frac{27a+9}{\left(3a-1\right)\left(3a+1\right)}.

$$\frac{9\left(3a+1\right)}{\left(3a-1\right)\left(3a+1\right)}$$

Cancel out 3a+1 in both numerator and denominator.

$$\frac{9}{3a-1}$$

Differentiate w.r.t. a

$$-\frac{27}{\left(3a-1\right)^{2}}$$