Solve w^2-3w-40/w^2-14w+48+w^2-w-30/w^2-3w-40

$$\frac { w ^ { 2 } - 3 w - 40 } { w ^ { 2 } - 14 w + 48 } + \frac { w ^ { 2 } - w - 30 } { w ^ { 2 } - 3 w - 40 }$$

Evaluate

$$\frac{2w^{2}-15w-4}{\left(w-8\right)\left(w-6\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{w^{2}-3w-40}{w^{2}-14w+48}.

$$\frac{\left(w-8\right)\left(w+5\right)}{\left(w-8\right)\left(w-6\right)}+\frac{w^{2}-w-30}{w^{2}-3w-40}$$

Cancel out w-8 in both numerator and denominator.

$$\frac{w+5}{w-6}+\frac{w^{2}-w-30}{w^{2}-3w-40}$$

Factor the expressions that are not already factored in \frac{w^{2}-w-30}{w^{2}-3w-40}.

$$\frac{w+5}{w-6}+\frac{\left(w-6\right)\left(w+5\right)}{\left(w-8\right)\left(w+5\right)}$$

Cancel out w+5 in both numerator and denominator.

$$\frac{w+5}{w-6}+\frac{w-6}{w-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of w-6 and w-8 is \left(w-8\right)\left(w-6\right). Multiply \frac{w+5}{w-6} times \frac{w-8}{w-8}. Multiply \frac{w-6}{w-8} times \frac{w-6}{w-6}.

$$\frac{\left(w+5\right)\left(w-8\right)}{\left(w-8\right)\left(w-6\right)}+\frac{\left(w-6\right)\left(w-6\right)}{\left(w-8\right)\left(w-6\right)}$$

Since \frac{\left(w+5\right)\left(w-8\right)}{\left(w-8\right)\left(w-6\right)} and \frac{\left(w-6\right)\left(w-6\right)}{\left(w-8\right)\left(w-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(w+5\right)\left(w-8\right)+\left(w-6\right)\left(w-6\right)}{\left(w-8\right)\left(w-6\right)}$$

Do the multiplications in \left(w+5\right)\left(w-8\right)+\left(w-6\right)\left(w-6\right).

$$\frac{w^{2}-8w+5w-40+w^{2}-6w-6w+36}{\left(w-8\right)\left(w-6\right)}$$

Combine like terms in w^{2}-8w+5w-40+w^{2}-6w-6w+36.

$$\frac{2w^{2}-15w-4}{\left(w-8\right)\left(w-6\right)}$$

Expand \left(w-8\right)\left(w-6\right).

$$\frac{2w^{2}-15w-4}{w^{2}-14w+48}$$

Expand

$$\frac{2w^{2}-15w-4}{\left(w-8\right)\left(w-6\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{w^{2}-3w-40}{w^{2}-14w+48}.

$$\frac{\left(w-8\right)\left(w+5\right)}{\left(w-8\right)\left(w-6\right)}+\frac{w^{2}-w-30}{w^{2}-3w-40}$$

Cancel out w-8 in both numerator and denominator.

$$\frac{w+5}{w-6}+\frac{w^{2}-w-30}{w^{2}-3w-40}$$

Factor the expressions that are not already factored in \frac{w^{2}-w-30}{w^{2}-3w-40}.

$$\frac{w+5}{w-6}+\frac{\left(w-6\right)\left(w+5\right)}{\left(w-8\right)\left(w+5\right)}$$

Cancel out w+5 in both numerator and denominator.

$$\frac{w+5}{w-6}+\frac{w-6}{w-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of w-6 and w-8 is \left(w-8\right)\left(w-6\right). Multiply \frac{w+5}{w-6} times \frac{w-8}{w-8}. Multiply \frac{w-6}{w-8} times \frac{w-6}{w-6}.

$$\frac{\left(w+5\right)\left(w-8\right)}{\left(w-8\right)\left(w-6\right)}+\frac{\left(w-6\right)\left(w-6\right)}{\left(w-8\right)\left(w-6\right)}$$

Since \frac{\left(w+5\right)\left(w-8\right)}{\left(w-8\right)\left(w-6\right)} and \frac{\left(w-6\right)\left(w-6\right)}{\left(w-8\right)\left(w-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(w+5\right)\left(w-8\right)+\left(w-6\right)\left(w-6\right)}{\left(w-8\right)\left(w-6\right)}$$

Do the multiplications in \left(w+5\right)\left(w-8\right)+\left(w-6\right)\left(w-6\right).

$$\frac{w^{2}-8w+5w-40+w^{2}-6w-6w+36}{\left(w-8\right)\left(w-6\right)}$$

Combine like terms in w^{2}-8w+5w-40+w^{2}-6w-6w+36.

$$\frac{2w^{2}-15w-4}{\left(w-8\right)\left(w-6\right)}$$

Expand \left(w-8\right)\left(w-6\right).

$$\frac{2w^{2}-15w-4}{w^{2}-14w+48}$$