Solve a-4/a-5-3/a+5-10/a^2-25

$$\frac { a - 4 } { a - 5 } - \frac { 3 } { a + 5 } - \frac { 10 } { a ^ { 2 } - 25 }$$

Evaluate

$$\frac{a+3}{a+5}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-5 and a+5 is \left(a-5\right)\left(a+5\right). Multiply \frac{a-4}{a-5} times \frac{a+5}{a+5}. Multiply \frac{3}{a+5} times \frac{a-5}{a-5}.

$$\frac{\left(a-4\right)\left(a+5\right)}{\left(a-5\right)\left(a+5\right)}-\frac{3\left(a-5\right)}{\left(a-5\right)\left(a+5\right)}-\frac{10}{a^{2}-25}$$

Since \frac{\left(a-4\right)\left(a+5\right)}{\left(a-5\right)\left(a+5\right)} and \frac{3\left(a-5\right)}{\left(a-5\right)\left(a+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a-4\right)\left(a+5\right)-3\left(a-5\right)}{\left(a-5\right)\left(a+5\right)}-\frac{10}{a^{2}-25}$$

Do the multiplications in \left(a-4\right)\left(a+5\right)-3\left(a-5\right).

$$\frac{a^{2}+5a-4a-20-3a+15}{\left(a-5\right)\left(a+5\right)}-\frac{10}{a^{2}-25}$$

Combine like terms in a^{2}+5a-4a-20-3a+15.

$$\frac{a^{2}-2a-5}{\left(a-5\right)\left(a+5\right)}-\frac{10}{a^{2}-25}$$

Factor a^{2}-25.

$$\frac{a^{2}-2a-5}{\left(a-5\right)\left(a+5\right)}-\frac{10}{\left(a-5\right)\left(a+5\right)}$$

Since \frac{a^{2}-2a-5}{\left(a-5\right)\left(a+5\right)} and \frac{10}{\left(a-5\right)\left(a+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a^{2}-2a-5-10}{\left(a-5\right)\left(a+5\right)}$$

Combine like terms in a^{2}-2a-5-10.

$$\frac{a^{2}-2a-15}{\left(a-5\right)\left(a+5\right)}$$

Factor the expressions that are not already factored in \frac{a^{2}-2a-15}{\left(a-5\right)\left(a+5\right)}.

$$\frac{\left(a-5\right)\left(a+3\right)}{\left(a-5\right)\left(a+5\right)}$$

Cancel out a-5 in both numerator and denominator.

$$\frac{a+3}{a+5}$$

Expand

$$\frac{a+3}{a+5}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-5 and a+5 is \left(a-5\right)\left(a+5\right). Multiply \frac{a-4}{a-5} times \frac{a+5}{a+5}. Multiply \frac{3}{a+5} times \frac{a-5}{a-5}.

$$\frac{\left(a-4\right)\left(a+5\right)}{\left(a-5\right)\left(a+5\right)}-\frac{3\left(a-5\right)}{\left(a-5\right)\left(a+5\right)}-\frac{10}{a^{2}-25}$$

Since \frac{\left(a-4\right)\left(a+5\right)}{\left(a-5\right)\left(a+5\right)} and \frac{3\left(a-5\right)}{\left(a-5\right)\left(a+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a-4\right)\left(a+5\right)-3\left(a-5\right)}{\left(a-5\right)\left(a+5\right)}-\frac{10}{a^{2}-25}$$

Do the multiplications in \left(a-4\right)\left(a+5\right)-3\left(a-5\right).

$$\frac{a^{2}+5a-4a-20-3a+15}{\left(a-5\right)\left(a+5\right)}-\frac{10}{a^{2}-25}$$

Combine like terms in a^{2}+5a-4a-20-3a+15.

$$\frac{a^{2}-2a-5}{\left(a-5\right)\left(a+5\right)}-\frac{10}{a^{2}-25}$$

Factor a^{2}-25.

$$\frac{a^{2}-2a-5}{\left(a-5\right)\left(a+5\right)}-\frac{10}{\left(a-5\right)\left(a+5\right)}$$

Since \frac{a^{2}-2a-5}{\left(a-5\right)\left(a+5\right)} and \frac{10}{\left(a-5\right)\left(a+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a^{2}-2a-5-10}{\left(a-5\right)\left(a+5\right)}$$

Combine like terms in a^{2}-2a-5-10.

$$\frac{a^{2}-2a-15}{\left(a-5\right)\left(a+5\right)}$$

Factor the expressions that are not already factored in \frac{a^{2}-2a-15}{\left(a-5\right)\left(a+5\right)}.

$$\frac{\left(a-5\right)\left(a+3\right)}{\left(a-5\right)\left(a+5\right)}$$

Cancel out a-5 in both numerator and denominator.

$$\frac{a+3}{a+5}$$