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

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

Evaluate

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

Short Solution Steps

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

$$\frac{a\left(-a+3\right)}{3\left(-a+3\right)}-\frac{3\left(a+2\right)}{3\left(-a+3\right)}-\frac{a^{2}+9}{3a-9}$$

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

$$\frac{a\left(-a+3\right)-3\left(a+2\right)}{3\left(-a+3\right)}-\frac{a^{2}+9}{3a-9}$$

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

$$\frac{-a^{2}+3a-3a-6}{3\left(-a+3\right)}-\frac{a^{2}+9}{3a-9}$$

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

$$\frac{-a^{2}-6}{3\left(-a+3\right)}-\frac{a^{2}+9}{3a-9}$$

Factor 3a-9.

$$\frac{-a^{2}-6}{3\left(-a+3\right)}-\frac{a^{2}+9}{3\left(a-3\right)}$$

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

$$\frac{-\left(-a^{2}-6\right)}{3\left(a-3\right)}-\frac{a^{2}+9}{3\left(a-3\right)}$$

Since \frac{-\left(-a^{2}-6\right)}{3\left(a-3\right)} and \frac{a^{2}+9}{3\left(a-3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(-a^{2}-6\right)-\left(a^{2}+9\right)}{3\left(a-3\right)}$$

Do the multiplications in -\left(-a^{2}-6\right)-\left(a^{2}+9\right).

$$\frac{a^{2}+6-a^{2}-9}{3\left(a-3\right)}$$

Combine like terms in a^{2}+6-a^{2}-9.

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

Expand 3\left(a-3\right).

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

Expand

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

Short Solution Steps

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

$$\frac{a\left(-a+3\right)}{3\left(-a+3\right)}-\frac{3\left(a+2\right)}{3\left(-a+3\right)}-\frac{a^{2}+9}{3a-9}$$

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

$$\frac{a\left(-a+3\right)-3\left(a+2\right)}{3\left(-a+3\right)}-\frac{a^{2}+9}{3a-9}$$

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

$$\frac{-a^{2}+3a-3a-6}{3\left(-a+3\right)}-\frac{a^{2}+9}{3a-9}$$

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

$$\frac{-a^{2}-6}{3\left(-a+3\right)}-\frac{a^{2}+9}{3a-9}$$

Factor 3a-9.

$$\frac{-a^{2}-6}{3\left(-a+3\right)}-\frac{a^{2}+9}{3\left(a-3\right)}$$

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

$$\frac{-\left(-a^{2}-6\right)}{3\left(a-3\right)}-\frac{a^{2}+9}{3\left(a-3\right)}$$

Since \frac{-\left(-a^{2}-6\right)}{3\left(a-3\right)} and \frac{a^{2}+9}{3\left(a-3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(-a^{2}-6\right)-\left(a^{2}+9\right)}{3\left(a-3\right)}$$

Do the multiplications in -\left(-a^{2}-6\right)-\left(a^{2}+9\right).

$$\frac{a^{2}+6-a^{2}-9}{3\left(a-3\right)}$$

Combine like terms in a^{2}+6-a^{2}-9.

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

Expand 3\left(a-3\right).

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