Solve a+4/a^2-6a+9:a^2-16/2a-6-2/a-4

$$\frac { a + 4 } { a ^ { 2 } - 6 a + 9 } : \frac { a ^ { 2 } - 16 } { 2 a - 6 } - \frac { 2 } { a - 4 }$$

Evaluate

$$-\frac{2}{a-3}$$

Short Solution Steps

Divide \frac{a+4}{a^{2}-6a+9} by \frac{a^{2}-16}{2a-6} by multiplying \frac{a+4}{a^{2}-6a+9} by the reciprocal of \frac{a^{2}-16}{2a-6}.

$$\frac{\left(a+4\right)\left(2a-6\right)}{\left(a^{2}-6a+9\right)\left(a^{2}-16\right)}-\frac{2}{a-4}$$

Factor the expressions that are not already factored in \frac{\left(a+4\right)\left(2a-6\right)}{\left(a^{2}-6a+9\right)\left(a^{2}-16\right)}.

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

Cancel out \left(a-3\right)\left(a+4\right) in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in 2-2a+6.

$$\frac{8-2a}{\left(a-4\right)\left(a-3\right)}$$

Factor the expressions that are not already factored in \frac{8-2a}{\left(a-4\right)\left(a-3\right)}.

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

Extract the negative sign in 4-a.

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

Cancel out a-4 in both numerator and denominator.

$$\frac{-2}{a-3}$$

Expand

$$-\frac{2}{a-3}$$

Short Solution Steps

Divide \frac{a+4}{a^{2}-6a+9} by \frac{a^{2}-16}{2a-6} by multiplying \frac{a+4}{a^{2}-6a+9} by the reciprocal of \frac{a^{2}-16}{2a-6}.

$$\frac{\left(a+4\right)\left(2a-6\right)}{\left(a^{2}-6a+9\right)\left(a^{2}-16\right)}-\frac{2}{a-4}$$

Factor the expressions that are not already factored in \frac{\left(a+4\right)\left(2a-6\right)}{\left(a^{2}-6a+9\right)\left(a^{2}-16\right)}.

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

Cancel out \left(a-3\right)\left(a+4\right) in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in 2-2a+6.

$$\frac{8-2a}{\left(a-4\right)\left(a-3\right)}$$

Factor the expressions that are not already factored in \frac{8-2a}{\left(a-4\right)\left(a-3\right)}.

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

Extract the negative sign in 4-a.

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

Cancel out a-4 in both numerator and denominator.

$$\frac{-2}{a-3}$$