Solve w+5/w^2+10w+25-w-8/w^2-10w+25

$$\frac { w + 5 } { w ^ { 2 } + 10 w + 25 } - \frac { w - 8 } { w ^ { 2 } - 10 w + 25 }$$

Evaluate

$$\frac{65-7w}{\left(w+5\right)\left(w-5\right)^{2}}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{w+5}{w^{2}+10w+25}.

$$\frac{w+5}{\left(w+5\right)^{2}}-\frac{w-8}{w^{2}-10w+25}$$

Cancel out w+5 in both numerator and denominator.

$$\frac{1}{w+5}-\frac{w-8}{w^{2}-10w+25}$$

Factor w^{2}-10w+25.

$$\frac{1}{w+5}-\frac{w-8}{\left(w-5\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of w+5 and \left(w-5\right)^{2} is \left(w+5\right)\left(w-5\right)^{2}. Multiply \frac{1}{w+5} times \frac{\left(w-5\right)^{2}}{\left(w-5\right)^{2}}. Multiply \frac{w-8}{\left(w-5\right)^{2}} times \frac{w+5}{w+5}.

$$\frac{\left(w-5\right)^{2}}{\left(w+5\right)\left(w-5\right)^{2}}-\frac{\left(w-8\right)\left(w+5\right)}{\left(w+5\right)\left(w-5\right)^{2}}$$

Since \frac{\left(w-5\right)^{2}}{\left(w+5\right)\left(w-5\right)^{2}} and \frac{\left(w-8\right)\left(w+5\right)}{\left(w+5\right)\left(w-5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(w-5\right)^{2}-\left(w-8\right)\left(w+5\right)}{\left(w+5\right)\left(w-5\right)^{2}}$$

Do the multiplications in \left(w-5\right)^{2}-\left(w-8\right)\left(w+5\right).

$$\frac{w^{2}-10w+25-w^{2}-5w+8w+40}{\left(w+5\right)\left(w-5\right)^{2}}$$

Combine like terms in w^{2}-10w+25-w^{2}-5w+8w+40.

$$\frac{-7w+65}{\left(w+5\right)\left(w-5\right)^{2}}$$

Expand \left(w+5\right)\left(w-5\right)^{2}.

$$\frac{-7w+65}{w^{3}-5w^{2}-25w+125}$$

Expand

$$\frac{65-7w}{\left(w+5\right)\left(w^{2}-10w+25\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{w+5}{w^{2}+10w+25}.

$$\frac{w+5}{\left(w+5\right)^{2}}-\frac{w-8}{w^{2}-10w+25}$$

Cancel out w+5 in both numerator and denominator.

$$\frac{1}{w+5}-\frac{w-8}{w^{2}-10w+25}$$

Factor w^{2}-10w+25.

$$\frac{1}{w+5}-\frac{w-8}{\left(w-5\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of w+5 and \left(w-5\right)^{2} is \left(w+5\right)\left(w-5\right)^{2}. Multiply \frac{1}{w+5} times \frac{\left(w-5\right)^{2}}{\left(w-5\right)^{2}}. Multiply \frac{w-8}{\left(w-5\right)^{2}} times \frac{w+5}{w+5}.

$$\frac{\left(w-5\right)^{2}}{\left(w+5\right)\left(w-5\right)^{2}}-\frac{\left(w-8\right)\left(w+5\right)}{\left(w+5\right)\left(w-5\right)^{2}}$$

Since \frac{\left(w-5\right)^{2}}{\left(w+5\right)\left(w-5\right)^{2}} and \frac{\left(w-8\right)\left(w+5\right)}{\left(w+5\right)\left(w-5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(w-5\right)^{2}-\left(w-8\right)\left(w+5\right)}{\left(w+5\right)\left(w-5\right)^{2}}$$

Do the multiplications in \left(w-5\right)^{2}-\left(w-8\right)\left(w+5\right).

$$\frac{w^{2}-10w+25-w^{2}-5w+8w+40}{\left(w+5\right)\left(w-5\right)^{2}}$$

Combine like terms in w^{2}-10w+25-w^{2}-5w+8w+40.

$$\frac{-7w+65}{\left(w+5\right)\left(w-5\right)^{2}}$$

Expand \left(w+5\right)\left(w-5\right)^{2}.

$$\frac{-7w+65}{w^{3}-5w^{2}-25w+125}$$