Solve 3a+2/36-a^2-a-4/a^2-8a+12

$$\frac { 3 a + 2 } { 36 - a ^ { 2 } } - \frac { a - 4 } { a ^ { 2 } - 8 a + 12 }$$

Evaluate

$$-\frac{2\left(2a^{2}-a-14\right)}{\left(a-2\right)\left(a^{2}-36\right)}$$

Short Solution Steps

Factor 36-a^{2}. Factor a^{2}-8a+12.

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

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

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

Since \frac{\left(3a+2\right)\left(a-2\right)}{\left(a-6\right)\left(a-2\right)\left(-a-6\right)} and \frac{\left(a-4\right)\left(-a-6\right)}{\left(a-6\right)\left(a-2\right)\left(-a-6\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3a+2\right)\left(a-2\right)-\left(a-4\right)\left(-a-6\right).

$$\frac{3a^{2}-6a+2a-4+a^{2}+6a-4a-24}{\left(a-6\right)\left(a-2\right)\left(-a-6\right)}$$

Combine like terms in 3a^{2}-6a+2a-4+a^{2}+6a-4a-24.

$$\frac{4a^{2}-2a-28}{\left(a-6\right)\left(a-2\right)\left(-a-6\right)}$$

Expand \left(a-6\right)\left(a-2\right)\left(-a-6\right).

$$\frac{4a^{2}-2a-28}{-a^{3}+2a^{2}+36a-72}$$

Expand

$$-\frac{2\left(2a^{2}-a-14\right)}{\left(a-2\right)\left(a^{2}-36\right)}$$

Short Solution Steps

Factor 36-a^{2}. Factor a^{2}-8a+12.

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

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

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

Since \frac{\left(3a+2\right)\left(a-2\right)}{\left(a-6\right)\left(a-2\right)\left(-a-6\right)} and \frac{\left(a-4\right)\left(-a-6\right)}{\left(a-6\right)\left(a-2\right)\left(-a-6\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3a+2\right)\left(a-2\right)-\left(a-4\right)\left(-a-6\right).

$$\frac{3a^{2}-6a+2a-4+a^{2}+6a-4a-24}{\left(a-6\right)\left(a-2\right)\left(-a-6\right)}$$

Combine like terms in 3a^{2}-6a+2a-4+a^{2}+6a-4a-24.

$$\frac{4a^{2}-2a-28}{\left(a-6\right)\left(a-2\right)\left(-a-6\right)}$$

Expand \left(a-6\right)\left(a-2\right)\left(-a-6\right).

$$\frac{4a^{2}-2a-28}{-a^{3}+2a^{2}+36a-72}$$