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

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{2a}{2a+6}.

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

Cancel out 2 in both numerator and denominator.

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

Factor a^{2}-9.

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

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

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

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

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

Do the multiplications in a\left(a-3\right)-\left(a^{2}+9\right).

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

Combine like terms in a^{2}-3a-a^{2}-9.

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

Factor the expressions that are not already factored in \frac{-3a-9}{\left(a-3\right)\left(a+3\right)}.

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

Extract the negative sign in -3-a.

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

Cancel out a+3 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{2a}{2a+6}.

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

Cancel out 2 in both numerator and denominator.

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

Factor a^{2}-9.

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

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

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

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

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

Do the multiplications in a\left(a-3\right)-\left(a^{2}+9\right).

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

Combine like terms in a^{2}-3a-a^{2}-9.

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

Factor the expressions that are not already factored in \frac{-3a-9}{\left(a-3\right)\left(a+3\right)}.

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

Extract the negative sign in -3-a.

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

Cancel out a+3 in both numerator and denominator.

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