Solve a-1/a+1*15a+11/2+3a/a-1

$$\frac { a - 1 } { a + 1 } \cdot 15 a + \frac { 11 } { 2 } + \frac { 3 a } { a - 1 }$$

Evaluate

$$\frac{\left(2a-1\right)\left(15a^{2}-14a+11\right)}{2\left(a^{2}-1\right)}$$

Short Solution Steps

Express \frac{a-1}{a+1}\times 15 as a single fraction.

$$\frac{\left(a-1\right)\times 15}{a+1}a+\frac{11}{2}+\frac{3a}{a-1}$$

Express \frac{\left(a-1\right)\times 15}{a+1}a as a single fraction.

$$\frac{\left(a-1\right)\times 15a}{a+1}+\frac{11}{2}+\frac{3a}{a-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and 2 is 2\left(a+1\right). Multiply \frac{\left(a-1\right)\times 15a}{a+1} times \frac{2}{2}. Multiply \frac{11}{2} times \frac{a+1}{a+1}.

$$\frac{2\left(a-1\right)\times 15a}{2\left(a+1\right)}+\frac{11\left(a+1\right)}{2\left(a+1\right)}+\frac{3a}{a-1}$$

Since \frac{2\left(a-1\right)\times 15a}{2\left(a+1\right)} and \frac{11\left(a+1\right)}{2\left(a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(a-1\right)\times 15a+11\left(a+1\right)}{2\left(a+1\right)}+\frac{3a}{a-1}$$

Do the multiplications in 2\left(a-1\right)\times 15a+11\left(a+1\right).

$$\frac{30a^{2}-30a+11a+11}{2\left(a+1\right)}+\frac{3a}{a-1}$$

Combine like terms in 30a^{2}-30a+11a+11.

$$\frac{30a^{2}-19a+11}{2\left(a+1\right)}+\frac{3a}{a-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(a+1\right) and a-1 is 2\left(a-1\right)\left(a+1\right). Multiply \frac{30a^{2}-19a+11}{2\left(a+1\right)} times \frac{a-1}{a-1}. Multiply \frac{3a}{a-1} times \frac{2\left(a+1\right)}{2\left(a+1\right)}.

$$\frac{\left(30a^{2}-19a+11\right)\left(a-1\right)}{2\left(a-1\right)\left(a+1\right)}+\frac{3a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)}$$

Since \frac{\left(30a^{2}-19a+11\right)\left(a-1\right)}{2\left(a-1\right)\left(a+1\right)} and \frac{3a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(30a^{2}-19a+11\right)\left(a-1\right)+3a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)}$$

Do the multiplications in \left(30a^{2}-19a+11\right)\left(a-1\right)+3a\times 2\left(a+1\right).

$$\frac{30a^{3}-30a^{2}-19a^{2}+19a+11a-11+6a^{2}+6a}{2\left(a-1\right)\left(a+1\right)}$$

Combine like terms in 30a^{3}-30a^{2}-19a^{2}+19a+11a-11+6a^{2}+6a.

$$\frac{30a^{3}-43a^{2}+36a-11}{2\left(a-1\right)\left(a+1\right)}$$

Expand 2\left(a-1\right)\left(a+1\right).

$$\frac{30a^{3}-43a^{2}+36a-11}{2a^{2}-2}$$

Expand

$$\frac{30a^{3}-43a^{2}+36a-11}{2\left(a^{2}-1\right)}$$

Short Solution Steps

Express \frac{a-1}{a+1}\times 15 as a single fraction.

$$\frac{\left(a-1\right)\times 15}{a+1}a+\frac{11}{2}+\frac{3a}{a-1}$$

Express \frac{\left(a-1\right)\times 15}{a+1}a as a single fraction.

$$\frac{\left(a-1\right)\times 15a}{a+1}+\frac{11}{2}+\frac{3a}{a-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and 2 is 2\left(a+1\right). Multiply \frac{\left(a-1\right)\times 15a}{a+1} times \frac{2}{2}. Multiply \frac{11}{2} times \frac{a+1}{a+1}.

$$\frac{2\left(a-1\right)\times 15a}{2\left(a+1\right)}+\frac{11\left(a+1\right)}{2\left(a+1\right)}+\frac{3a}{a-1}$$

Since \frac{2\left(a-1\right)\times 15a}{2\left(a+1\right)} and \frac{11\left(a+1\right)}{2\left(a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(a-1\right)\times 15a+11\left(a+1\right)}{2\left(a+1\right)}+\frac{3a}{a-1}$$

Do the multiplications in 2\left(a-1\right)\times 15a+11\left(a+1\right).

$$\frac{30a^{2}-30a+11a+11}{2\left(a+1\right)}+\frac{3a}{a-1}$$

Combine like terms in 30a^{2}-30a+11a+11.

$$\frac{30a^{2}-19a+11}{2\left(a+1\right)}+\frac{3a}{a-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(a+1\right) and a-1 is 2\left(a-1\right)\left(a+1\right). Multiply \frac{30a^{2}-19a+11}{2\left(a+1\right)} times \frac{a-1}{a-1}. Multiply \frac{3a}{a-1} times \frac{2\left(a+1\right)}{2\left(a+1\right)}.

$$\frac{\left(30a^{2}-19a+11\right)\left(a-1\right)}{2\left(a-1\right)\left(a+1\right)}+\frac{3a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)}$$

Since \frac{\left(30a^{2}-19a+11\right)\left(a-1\right)}{2\left(a-1\right)\left(a+1\right)} and \frac{3a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(30a^{2}-19a+11\right)\left(a-1\right)+3a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)}$$

Do the multiplications in \left(30a^{2}-19a+11\right)\left(a-1\right)+3a\times 2\left(a+1\right).

$$\frac{30a^{3}-30a^{2}-19a^{2}+19a+11a-11+6a^{2}+6a}{2\left(a-1\right)\left(a+1\right)}$$

Combine like terms in 30a^{3}-30a^{2}-19a^{2}+19a+11a-11+6a^{2}+6a.

$$\frac{30a^{3}-43a^{2}+36a-11}{2\left(a-1\right)\left(a+1\right)}$$

Expand 2\left(a-1\right)\left(a+1\right).

$$\frac{30a^{3}-43a^{2}+36a-11}{2a^{2}-2}$$