Solve m-2/m-5:(m^2+24/m^2-25-4/m-5)

$$\frac { m - 2 } { m - 5 } : ( \frac { m ^ { 2 } + 24 } { m ^ { 2 } - 25 } - \frac { 4 } { m - 5 } )$$

Evaluate

$$\frac{m+5}{m-2}$$

Short Solution Steps

Factor m^{2}-25.

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+24}{\left(m-5\right)\left(m+5\right)}-\frac{4}{m-5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(m-5\right)\left(m+5\right) and m-5 is \left(m-5\right)\left(m+5\right). Multiply \frac{4}{m-5} times \frac{m+5}{m+5}.

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+24}{\left(m-5\right)\left(m+5\right)}-\frac{4\left(m+5\right)}{\left(m-5\right)\left(m+5\right)}}$$

Since \frac{m^{2}+24}{\left(m-5\right)\left(m+5\right)} and \frac{4\left(m+5\right)}{\left(m-5\right)\left(m+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+24-4\left(m+5\right)}{\left(m-5\right)\left(m+5\right)}}$$

Do the multiplications in m^{2}+24-4\left(m+5\right).

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+24-4m-20}{\left(m-5\right)\left(m+5\right)}}$$

Combine like terms in m^{2}+24-4m-20.

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+4-4m}{\left(m-5\right)\left(m+5\right)}}$$

Divide \frac{m-2}{m-5} by \frac{m^{2}+4-4m}{\left(m-5\right)\left(m+5\right)} by multiplying \frac{m-2}{m-5} by the reciprocal of \frac{m^{2}+4-4m}{\left(m-5\right)\left(m+5\right)}.

$$\frac{\left(m-2\right)\left(m-5\right)\left(m+5\right)}{\left(m-5\right)\left(m^{2}+4-4m\right)}$$

Cancel out m-5 in both numerator and denominator.

$$\frac{\left(m-2\right)\left(m+5\right)}{m^{2}-4m+4}$$

Factor the expressions that are not already factored.

$$\frac{\left(m-2\right)\left(m+5\right)}{\left(m-2\right)^{2}}$$

Cancel out m-2 in both numerator and denominator.

$$\frac{m+5}{m-2}$$

Expand

$$\frac{m+5}{m-2}$$

Short Solution Steps

Factor m^{2}-25.

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+24}{\left(m-5\right)\left(m+5\right)}-\frac{4}{m-5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(m-5\right)\left(m+5\right) and m-5 is \left(m-5\right)\left(m+5\right). Multiply \frac{4}{m-5} times \frac{m+5}{m+5}.

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+24}{\left(m-5\right)\left(m+5\right)}-\frac{4\left(m+5\right)}{\left(m-5\right)\left(m+5\right)}}$$

Since \frac{m^{2}+24}{\left(m-5\right)\left(m+5\right)} and \frac{4\left(m+5\right)}{\left(m-5\right)\left(m+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+24-4\left(m+5\right)}{\left(m-5\right)\left(m+5\right)}}$$

Do the multiplications in m^{2}+24-4\left(m+5\right).

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+24-4m-20}{\left(m-5\right)\left(m+5\right)}}$$

Combine like terms in m^{2}+24-4m-20.

$$\frac{\frac{m-2}{m-5}}{\frac{m^{2}+4-4m}{\left(m-5\right)\left(m+5\right)}}$$

Divide \frac{m-2}{m-5} by \frac{m^{2}+4-4m}{\left(m-5\right)\left(m+5\right)} by multiplying \frac{m-2}{m-5} by the reciprocal of \frac{m^{2}+4-4m}{\left(m-5\right)\left(m+5\right)}.

$$\frac{\left(m-2\right)\left(m-5\right)\left(m+5\right)}{\left(m-5\right)\left(m^{2}+4-4m\right)}$$

Cancel out m-5 in both numerator and denominator.

$$\frac{\left(m-2\right)\left(m+5\right)}{m^{2}-4m+4}$$

Factor the expressions that are not already factored.

$$\frac{\left(m-2\right)\left(m+5\right)}{\left(m-2\right)^{2}}$$

Cancel out m-2 in both numerator and denominator.

$$\frac{m+5}{m-2}$$