Solve -a/a^2-9+1/3a+9+1/2a-6

$$\frac { - a } { a ^ { 2 } - 9 } + \frac { 1 } { 3 a + 9 } + \frac { 1 } { 2 a - 6 }$$

Evaluate

$$-\frac{1}{6\left(a+3\right)}$$

Short Solution Steps

Factor a^{2}-9. Factor 3a+9.

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

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

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

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

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

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

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

Combine like terms in -3a+a-3.

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

Factor 2a-6.

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

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

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

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

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

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

$$\frac{-4a-6+3a+9}{6\left(a-3\right)\left(a+3\right)}$$

Combine like terms in -4a-6+3a+9.

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

Extract the negative sign in -a+3.

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

Cancel out a-3 in both numerator and denominator.

$$\frac{-1}{6\left(a+3\right)}$$

Expand 6\left(a+3\right).

$$\frac{-1}{6a+18}$$

Factor

$$-\frac{1}{6\left(a+3\right)}$$