Solve frac{x_{0}+h+1}{x_{0}+h-2}-frac{x_{0}+h}{x_{0}-2}

$$\frac { x _ { 0 } + h + 1 } { x _ { 0 } + h - 2 } - \frac { x _ { 0 } + h } { x _ { 0 } - 2 }$$

Evaluate

$$\frac{-hx_{0}+x_{0}-h^{2}-2}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x_{0}+h-2 and x_{0}-2 is \left(x_{0}-2\right)\left(x_{0}+h-2\right). Multiply \frac{x_{0}+h+1}{x_{0}+h-2} times \frac{x_{0}-2}{x_{0}-2}. Multiply \frac{x_{0}+h}{x_{0}-2} times \frac{x_{0}+h-2}{x_{0}+h-2}.

$$\frac{\left(x_{0}+h+1\right)\left(x_{0}-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}-\frac{\left(x_{0}+h\right)\left(x_{0}+h-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Since \frac{\left(x_{0}+h+1\right)\left(x_{0}-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)} and \frac{\left(x_{0}+h\right)\left(x_{0}+h-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x_{0}+h+1\right)\left(x_{0}-2\right)-\left(x_{0}+h\right)\left(x_{0}+h-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Do the multiplications in \left(x_{0}+h+1\right)\left(x_{0}-2\right)-\left(x_{0}+h\right)\left(x_{0}+h-2\right).

$$\frac{x_{0}^{2}-2x_{0}+hx_{0}-2h+x_{0}-2-x_{0}^{2}-x_{0}h+2x_{0}-x_{0}h-h^{2}+2h}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Combine like terms in x_{0}^{2}-2x_{0}+hx_{0}-2h+x_{0}-2-x_{0}^{2}-x_{0}h+2x_{0}-x_{0}h-h^{2}+2h.

$$\frac{x_{0}-hx_{0}-2-h^{2}}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Expand \left(x_{0}-2\right)\left(x_{0}+h-2\right).

$$\frac{x_{0}-hx_{0}-2-h^{2}}{x_{0}^{2}+hx_{0}-4x_{0}-2h+4}$$

Expand

$$\frac{-hx_{0}+x_{0}-h^{2}-2}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x_{0}+h-2 and x_{0}-2 is \left(x_{0}-2\right)\left(x_{0}+h-2\right). Multiply \frac{x_{0}+h+1}{x_{0}+h-2} times \frac{x_{0}-2}{x_{0}-2}. Multiply \frac{x_{0}+h}{x_{0}-2} times \frac{x_{0}+h-2}{x_{0}+h-2}.

$$\frac{\left(x_{0}+h+1\right)\left(x_{0}-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}-\frac{\left(x_{0}+h\right)\left(x_{0}+h-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Since \frac{\left(x_{0}+h+1\right)\left(x_{0}-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)} and \frac{\left(x_{0}+h\right)\left(x_{0}+h-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x_{0}+h+1\right)\left(x_{0}-2\right)-\left(x_{0}+h\right)\left(x_{0}+h-2\right)}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Do the multiplications in \left(x_{0}+h+1\right)\left(x_{0}-2\right)-\left(x_{0}+h\right)\left(x_{0}+h-2\right).

$$\frac{x_{0}^{2}-2x_{0}+hx_{0}-2h+x_{0}-2-x_{0}^{2}-x_{0}h+2x_{0}-x_{0}h-h^{2}+2h}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Combine like terms in x_{0}^{2}-2x_{0}+hx_{0}-2h+x_{0}-2-x_{0}^{2}-x_{0}h+2x_{0}-x_{0}h-h^{2}+2h.

$$\frac{x_{0}-hx_{0}-2-h^{2}}{\left(x_{0}-2\right)\left(x_{0}+h-2\right)}$$

Expand \left(x_{0}-2\right)\left(x_{0}+h-2\right).

$$\frac{x_{0}-hx_{0}-2-h^{2}}{x_{0}^{2}+hx_{0}-4x_{0}-2h+4}$$