Solve x-15/x^2-10/x^2-100

$$\frac { x - 15 } { x ^ { 2 } } - \frac { 10 } { x ^ { 2 } - 100 }$$

Evaluate

$$\frac{x^{3}-25x^{2}-100x+1500}{x^{2}\left(x^{2}-100\right)}$$

Short Solution Steps

Factor x^{2}-100.

$$\frac{x-15}{x^{2}}-\frac{10}{\left(x-10\right)\left(x+10\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and \left(x-10\right)\left(x+10\right) is \left(x-10\right)\left(x+10\right)x^{2}. Multiply \frac{x-15}{x^{2}} times \frac{\left(x-10\right)\left(x+10\right)}{\left(x-10\right)\left(x+10\right)}. Multiply \frac{10}{\left(x-10\right)\left(x+10\right)} times \frac{x^{2}}{x^{2}}.

$$\frac{\left(x-15\right)\left(x-10\right)\left(x+10\right)}{\left(x-10\right)\left(x+10\right)x^{2}}-\frac{10x^{2}}{\left(x-10\right)\left(x+10\right)x^{2}}$$

Since \frac{\left(x-15\right)\left(x-10\right)\left(x+10\right)}{\left(x-10\right)\left(x+10\right)x^{2}} and \frac{10x^{2}}{\left(x-10\right)\left(x+10\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-15\right)\left(x-10\right)\left(x+10\right)-10x^{2}}{\left(x-10\right)\left(x+10\right)x^{2}}$$

Do the multiplications in \left(x-15\right)\left(x-10\right)\left(x+10\right)-10x^{2}.

$$\frac{x^{3}-100x-15x^{2}+1500-10x^{2}}{\left(x-10\right)\left(x+10\right)x^{2}}$$

Combine like terms in x^{3}-100x-15x^{2}+1500-10x^{2}.

$$\frac{x^{3}-100x-25x^{2}+1500}{\left(x-10\right)\left(x+10\right)x^{2}}$$

Expand \left(x-10\right)\left(x+10\right)x^{2}.

$$\frac{x^{3}-100x-25x^{2}+1500}{x^{4}-100x^{2}}$$

Expand

$$\frac{x^{3}-25x^{2}-100x+1500}{x^{2}\left(x^{2}-100\right)}$$

Short Solution Steps

Factor x^{2}-100.

$$\frac{x-15}{x^{2}}-\frac{10}{\left(x-10\right)\left(x+10\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and \left(x-10\right)\left(x+10\right) is \left(x-10\right)\left(x+10\right)x^{2}. Multiply \frac{x-15}{x^{2}} times \frac{\left(x-10\right)\left(x+10\right)}{\left(x-10\right)\left(x+10\right)}. Multiply \frac{10}{\left(x-10\right)\left(x+10\right)} times \frac{x^{2}}{x^{2}}.

$$\frac{\left(x-15\right)\left(x-10\right)\left(x+10\right)}{\left(x-10\right)\left(x+10\right)x^{2}}-\frac{10x^{2}}{\left(x-10\right)\left(x+10\right)x^{2}}$$

Since \frac{\left(x-15\right)\left(x-10\right)\left(x+10\right)}{\left(x-10\right)\left(x+10\right)x^{2}} and \frac{10x^{2}}{\left(x-10\right)\left(x+10\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-15\right)\left(x-10\right)\left(x+10\right)-10x^{2}}{\left(x-10\right)\left(x+10\right)x^{2}}$$

Do the multiplications in \left(x-15\right)\left(x-10\right)\left(x+10\right)-10x^{2}.

$$\frac{x^{3}-100x-15x^{2}+1500-10x^{2}}{\left(x-10\right)\left(x+10\right)x^{2}}$$

Combine like terms in x^{3}-100x-15x^{2}+1500-10x^{2}.

$$\frac{x^{3}-100x-25x^{2}+1500}{\left(x-10\right)\left(x+10\right)x^{2}}$$

Expand \left(x-10\right)\left(x+10\right)x^{2}.

$$\frac{x^{3}-100x-25x^{2}+1500}{x^{4}-100x^{2}}$$