Solve x+h-1/x+h-x+1/x/h

$$\frac{ x+h- \frac{ 1 }{ x+h } -x+ \frac{ 1 }{ x } }{ h }$$

Evaluate

$$\frac{x^{2}+hx+1}{x\left(x+h\right)}$$

Short Solution Steps

Combine x and -x to get 0.

$$\frac{h-\frac{1}{x+h}+\frac{1}{x}}{h}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply h times \frac{x+h}{x+h}.

$$\frac{\frac{h\left(x+h\right)}{x+h}-\frac{1}{x+h}+\frac{1}{x}}{h}$$

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

$$\frac{\frac{h\left(x+h\right)-1}{x+h}+\frac{1}{x}}{h}$$

Do the multiplications in h\left(x+h\right)-1.

$$\frac{\frac{hx+h^{2}-1}{x+h}+\frac{1}{x}}{h}$$

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

$$\frac{\frac{\left(hx+h^{2}-1\right)x}{x\left(x+h\right)}+\frac{x+h}{x\left(x+h\right)}}{h}$$

Since \frac{\left(hx+h^{2}-1\right)x}{x\left(x+h\right)} and \frac{x+h}{x\left(x+h\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(hx+h^{2}-1\right)x+x+h}{x\left(x+h\right)}}{h}$$

Do the multiplications in \left(hx+h^{2}-1\right)x+x+h.

$$\frac{\frac{hx^{2}+h^{2}x-x+x+h}{x\left(x+h\right)}}{h}$$

Combine like terms in hx^{2}+h^{2}x-x+x+h.

$$\frac{\frac{h+h^{2}x+hx^{2}}{x\left(x+h\right)}}{h}$$

Express \frac{\frac{h+h^{2}x+hx^{2}}{x\left(x+h\right)}}{h} as a single fraction.

$$\frac{h+h^{2}x+hx^{2}}{x\left(x+h\right)h}$$

Factor the expressions that are not already factored.

$$\frac{h\left(x^{2}+hx+1\right)}{hx\left(x+h\right)}$$

Cancel out h in both numerator and denominator.

$$\frac{x^{2}+hx+1}{x\left(x+h\right)}$$

Expand the expression.

$$\frac{x^{2}+hx+1}{x^{2}+hx}$$

Expand

$$\frac{x^{2}+hx+1}{x\left(x+h\right)}$$

Short Solution Steps

Combine x and -x to get 0.

$$\frac{h-\frac{1}{x+h}+\frac{1}{x}}{h}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply h times \frac{x+h}{x+h}.

$$\frac{\frac{h\left(x+h\right)}{x+h}-\frac{1}{x+h}+\frac{1}{x}}{h}$$

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

$$\frac{\frac{h\left(x+h\right)-1}{x+h}+\frac{1}{x}}{h}$$

Do the multiplications in h\left(x+h\right)-1.

$$\frac{\frac{hx+h^{2}-1}{x+h}+\frac{1}{x}}{h}$$

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

$$\frac{\frac{\left(hx+h^{2}-1\right)x}{x\left(x+h\right)}+\frac{x+h}{x\left(x+h\right)}}{h}$$

Since \frac{\left(hx+h^{2}-1\right)x}{x\left(x+h\right)} and \frac{x+h}{x\left(x+h\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(hx+h^{2}-1\right)x+x+h}{x\left(x+h\right)}}{h}$$

Do the multiplications in \left(hx+h^{2}-1\right)x+x+h.

$$\frac{\frac{hx^{2}+h^{2}x-x+x+h}{x\left(x+h\right)}}{h}$$

Combine like terms in hx^{2}+h^{2}x-x+x+h.

$$\frac{\frac{h+h^{2}x+hx^{2}}{x\left(x+h\right)}}{h}$$

Express \frac{\frac{h+h^{2}x+hx^{2}}{x\left(x+h\right)}}{h} as a single fraction.

$$\frac{h+h^{2}x+hx^{2}}{x\left(x+h\right)h}$$

Factor the expressions that are not already factored.

$$\frac{h\left(x^{2}+hx+1\right)}{hx\left(x+h\right)}$$

Cancel out h in both numerator and denominator.

$$\frac{x^{2}+hx+1}{x\left(x+h\right)}$$

Expand the expression.

$$\frac{x^{2}+hx+1}{x^{2}+hx}$$