Solve a-6/4a^2+4a+3a-7/2a^2-2a-a-7/a^2-1

$$\frac { a - 6 } { 4 a ^ { 2 } + 4 a } + \frac { 3 a - 7 } { 2 a ^ { 2 } - 2 a } - \frac { a - 7 } { a ^ { 2 } - 1 }$$

Evaluate

$$\frac{3a^{2}+13a-8}{4a\left(a^{2}-1\right)}$$

Short Solution Steps

Factor 4a^{2}+4a. Factor 2a^{2}-2a.

$$\frac{a-6}{4a\left(a+1\right)}+\frac{3a-7}{2a\left(a-1\right)}-\frac{a-7}{a^{2}-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4a\left(a+1\right) and 2a\left(a-1\right) is 4a\left(a-1\right)\left(a+1\right). Multiply \frac{a-6}{4a\left(a+1\right)} times \frac{a-1}{a-1}. Multiply \frac{3a-7}{2a\left(a-1\right)} times \frac{2\left(a+1\right)}{2\left(a+1\right)}.

$$\frac{\left(a-6\right)\left(a-1\right)}{4a\left(a-1\right)\left(a+1\right)}+\frac{\left(3a-7\right)\times 2\left(a+1\right)}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{a^{2}-1}$$

Since \frac{\left(a-6\right)\left(a-1\right)}{4a\left(a-1\right)\left(a+1\right)} and \frac{\left(3a-7\right)\times 2\left(a+1\right)}{4a\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a-6\right)\left(a-1\right)+\left(3a-7\right)\times 2\left(a+1\right)}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{a^{2}-1}$$

Do the multiplications in \left(a-6\right)\left(a-1\right)+\left(3a-7\right)\times 2\left(a+1\right).

$$\frac{a^{2}-a-6a+6+6a^{2}+6a-14a-14}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{a^{2}-1}$$

Combine like terms in a^{2}-a-6a+6+6a^{2}+6a-14a-14.

$$\frac{7a^{2}-15a-8}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{a^{2}-1}$$

Factor a^{2}-1.

$$\frac{7a^{2}-15a-8}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{\left(a-1\right)\left(a+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4a\left(a-1\right)\left(a+1\right) and \left(a-1\right)\left(a+1\right) is 4a\left(a-1\right)\left(a+1\right). Multiply \frac{a-7}{\left(a-1\right)\left(a+1\right)} times \frac{4a}{4a}.

$$\frac{7a^{2}-15a-8}{4a\left(a-1\right)\left(a+1\right)}-\frac{\left(a-7\right)\times 4a}{4a\left(a-1\right)\left(a+1\right)}$$

Since \frac{7a^{2}-15a-8}{4a\left(a-1\right)\left(a+1\right)} and \frac{\left(a-7\right)\times 4a}{4a\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7a^{2}-15a-8-\left(a-7\right)\times 4a}{4a\left(a-1\right)\left(a+1\right)}$$

Do the multiplications in 7a^{2}-15a-8-\left(a-7\right)\times 4a.

$$\frac{7a^{2}-15a-8-4a^{2}+28a}{4a\left(a-1\right)\left(a+1\right)}$$

Combine like terms in 7a^{2}-15a-8-4a^{2}+28a.

$$\frac{3a^{2}+13a-8}{4a\left(a-1\right)\left(a+1\right)}$$

Expand 4a\left(a-1\right)\left(a+1\right).

$$\frac{3a^{2}+13a-8}{4a^{3}-4a}$$

Expand

$$\frac{3a^{2}+13a-8}{4a\left(a^{2}-1\right)}$$

Short Solution Steps

Factor 4a^{2}+4a. Factor 2a^{2}-2a.

$$\frac{a-6}{4a\left(a+1\right)}+\frac{3a-7}{2a\left(a-1\right)}-\frac{a-7}{a^{2}-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4a\left(a+1\right) and 2a\left(a-1\right) is 4a\left(a-1\right)\left(a+1\right). Multiply \frac{a-6}{4a\left(a+1\right)} times \frac{a-1}{a-1}. Multiply \frac{3a-7}{2a\left(a-1\right)} times \frac{2\left(a+1\right)}{2\left(a+1\right)}.

$$\frac{\left(a-6\right)\left(a-1\right)}{4a\left(a-1\right)\left(a+1\right)}+\frac{\left(3a-7\right)\times 2\left(a+1\right)}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{a^{2}-1}$$

Since \frac{\left(a-6\right)\left(a-1\right)}{4a\left(a-1\right)\left(a+1\right)} and \frac{\left(3a-7\right)\times 2\left(a+1\right)}{4a\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a-6\right)\left(a-1\right)+\left(3a-7\right)\times 2\left(a+1\right)}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{a^{2}-1}$$

Do the multiplications in \left(a-6\right)\left(a-1\right)+\left(3a-7\right)\times 2\left(a+1\right).

$$\frac{a^{2}-a-6a+6+6a^{2}+6a-14a-14}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{a^{2}-1}$$

Combine like terms in a^{2}-a-6a+6+6a^{2}+6a-14a-14.

$$\frac{7a^{2}-15a-8}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{a^{2}-1}$$

Factor a^{2}-1.

$$\frac{7a^{2}-15a-8}{4a\left(a-1\right)\left(a+1\right)}-\frac{a-7}{\left(a-1\right)\left(a+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4a\left(a-1\right)\left(a+1\right) and \left(a-1\right)\left(a+1\right) is 4a\left(a-1\right)\left(a+1\right). Multiply \frac{a-7}{\left(a-1\right)\left(a+1\right)} times \frac{4a}{4a}.

$$\frac{7a^{2}-15a-8}{4a\left(a-1\right)\left(a+1\right)}-\frac{\left(a-7\right)\times 4a}{4a\left(a-1\right)\left(a+1\right)}$$

Since \frac{7a^{2}-15a-8}{4a\left(a-1\right)\left(a+1\right)} and \frac{\left(a-7\right)\times 4a}{4a\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7a^{2}-15a-8-\left(a-7\right)\times 4a}{4a\left(a-1\right)\left(a+1\right)}$$

Do the multiplications in 7a^{2}-15a-8-\left(a-7\right)\times 4a.

$$\frac{7a^{2}-15a-8-4a^{2}+28a}{4a\left(a-1\right)\left(a+1\right)}$$

Combine like terms in 7a^{2}-15a-8-4a^{2}+28a.

$$\frac{3a^{2}+13a-8}{4a\left(a-1\right)\left(a+1\right)}$$

Expand 4a\left(a-1\right)\left(a+1\right).

$$\frac{3a^{2}+13a-8}{4a^{3}-4a}$$