Solve -4/x-5+7x-27/x^2-8x+15

$$\frac { - 4 } { x - 5 } + \frac { 7 x - 27 } { x ^ { 2 } - 8 x + 15 }$$

Evaluate

$$\frac{3}{x-3}$$

Short Solution Steps

Factor x^{2}-8x+15.

$$\frac{-4}{x-5}+\frac{7x-27}{\left(x-5\right)\left(x-3\right)}$$

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

$$\frac{-4\left(x-3\right)}{\left(x-5\right)\left(x-3\right)}+\frac{7x-27}{\left(x-5\right)\left(x-3\right)}$$

Since \frac{-4\left(x-3\right)}{\left(x-5\right)\left(x-3\right)} and \frac{7x-27}{\left(x-5\right)\left(x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{-4\left(x-3\right)+7x-27}{\left(x-5\right)\left(x-3\right)}$$

Do the multiplications in -4\left(x-3\right)+7x-27.

$$\frac{-4x+12+7x-27}{\left(x-5\right)\left(x-3\right)}$$

Combine like terms in -4x+12+7x-27.

$$\frac{3x-15}{\left(x-5\right)\left(x-3\right)}$$

Factor the expressions that are not already factored in \frac{3x-15}{\left(x-5\right)\left(x-3\right)}.

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

Cancel out x-5 in both numerator and denominator.

$$\frac{3}{x-3}$$

Expand

$$\frac{3}{x-3}$$

Short Solution Steps

Factor x^{2}-8x+15.

$$\frac{-4}{x-5}+\frac{7x-27}{\left(x-5\right)\left(x-3\right)}$$

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

$$\frac{-4\left(x-3\right)}{\left(x-5\right)\left(x-3\right)}+\frac{7x-27}{\left(x-5\right)\left(x-3\right)}$$

Since \frac{-4\left(x-3\right)}{\left(x-5\right)\left(x-3\right)} and \frac{7x-27}{\left(x-5\right)\left(x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{-4\left(x-3\right)+7x-27}{\left(x-5\right)\left(x-3\right)}$$

Do the multiplications in -4\left(x-3\right)+7x-27.

$$\frac{-4x+12+7x-27}{\left(x-5\right)\left(x-3\right)}$$

Combine like terms in -4x+12+7x-27.

$$\frac{3x-15}{\left(x-5\right)\left(x-3\right)}$$

Factor the expressions that are not already factored in \frac{3x-15}{\left(x-5\right)\left(x-3\right)}.

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

Cancel out x-5 in both numerator and denominator.

$$\frac{3}{x-3}$$