Solve x-3/x^2-13x+40-9/x^2-7x+10

$$\frac { x - 3 } { x ^ { 2 } - 13 x + 40 } - \frac { 9 } { x ^ { 2 } - 7 x + 10 }$$

Evaluate

$$\frac{x^{2}-14x+78}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Short Solution Steps

Factor x^{2}-13x+40. Factor x^{2}-7x+10.

$$\frac{x-3}{\left(x-8\right)\left(x-5\right)}-\frac{9}{\left(x-5\right)\left(x-2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-8\right)\left(x-5\right) and \left(x-5\right)\left(x-2\right) is \left(x-8\right)\left(x-5\right)\left(x-2\right). Multiply \frac{x-3}{\left(x-8\right)\left(x-5\right)} times \frac{x-2}{x-2}. Multiply \frac{9}{\left(x-5\right)\left(x-2\right)} times \frac{x-8}{x-8}.

$$\frac{\left(x-3\right)\left(x-2\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}-\frac{9\left(x-8\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Since \frac{\left(x-3\right)\left(x-2\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)} and \frac{9\left(x-8\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-3\right)\left(x-2\right)-9\left(x-8\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Do the multiplications in \left(x-3\right)\left(x-2\right)-9\left(x-8\right).

$$\frac{x^{2}-2x-3x+6-9x+72}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Combine like terms in x^{2}-2x-3x+6-9x+72.

$$\frac{x^{2}-14x+78}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Expand \left(x-8\right)\left(x-5\right)\left(x-2\right).

$$\frac{x^{2}-14x+78}{x^{3}-15x^{2}+66x-80}$$

Expand

$$\frac{x^{2}-14x+78}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Short Solution Steps

Factor x^{2}-13x+40. Factor x^{2}-7x+10.

$$\frac{x-3}{\left(x-8\right)\left(x-5\right)}-\frac{9}{\left(x-5\right)\left(x-2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-8\right)\left(x-5\right) and \left(x-5\right)\left(x-2\right) is \left(x-8\right)\left(x-5\right)\left(x-2\right). Multiply \frac{x-3}{\left(x-8\right)\left(x-5\right)} times \frac{x-2}{x-2}. Multiply \frac{9}{\left(x-5\right)\left(x-2\right)} times \frac{x-8}{x-8}.

$$\frac{\left(x-3\right)\left(x-2\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}-\frac{9\left(x-8\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Since \frac{\left(x-3\right)\left(x-2\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)} and \frac{9\left(x-8\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-3\right)\left(x-2\right)-9\left(x-8\right)}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Do the multiplications in \left(x-3\right)\left(x-2\right)-9\left(x-8\right).

$$\frac{x^{2}-2x-3x+6-9x+72}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Combine like terms in x^{2}-2x-3x+6-9x+72.

$$\frac{x^{2}-14x+78}{\left(x-8\right)\left(x-5\right)\left(x-2\right)}$$

Expand \left(x-8\right)\left(x-5\right)\left(x-2\right).

$$\frac{x^{2}-14x+78}{x^{3}-15x^{2}+66x-80}$$