Solve (x/x^2-25-x-8/x^2-10x+25):x-20/(x-5)^2

$$( \frac { x } { x ^ { 2 } - 25 } - \frac { x - 8 } { x ^ { 2 } - 10 x + 25 } ) : \frac { x - 20 } { ( x - 5 ) ^ { 2 } }$$

Evaluate

$$-\frac{2}{x+5}$$

Short Solution Steps

Factor x^{2}-25. Factor x^{2}-10x+25.

$$\frac{\frac{x}{\left(x-5\right)\left(x+5\right)}-\frac{x-8}{\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

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

$$\frac{\frac{x\left(x-5\right)}{\left(x+5\right)\left(x-5\right)^{2}}-\frac{\left(x-8\right)\left(x+5\right)}{\left(x+5\right)\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

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

$$\frac{\frac{x\left(x-5\right)-\left(x-8\right)\left(x+5\right)}{\left(x+5\right)\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

Do the multiplications in x\left(x-5\right)-\left(x-8\right)\left(x+5\right).

$$\frac{\frac{x^{2}-5x-x^{2}-5x+8x+40}{\left(x+5\right)\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

Combine like terms in x^{2}-5x-x^{2}-5x+8x+40.

$$\frac{\frac{-2x+40}{\left(x+5\right)\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

Divide \frac{-2x+40}{\left(x+5\right)\left(x-5\right)^{2}} by \frac{x-20}{\left(x-5\right)^{2}} by multiplying \frac{-2x+40}{\left(x+5\right)\left(x-5\right)^{2}} by the reciprocal of \frac{x-20}{\left(x-5\right)^{2}}.

$$\frac{\left(-2x+40\right)\left(x-5\right)^{2}}{\left(x+5\right)\left(x-5\right)^{2}\left(x-20\right)}$$

Cancel out \left(x-5\right)^{2} in both numerator and denominator.

$$\frac{-2x+40}{\left(x-20\right)\left(x+5\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\left(-x+20\right)}{\left(x-20\right)\left(x+5\right)}$$

Extract the negative sign in 20-x.

$$\frac{-2\left(x-20\right)}{\left(x-20\right)\left(x+5\right)}$$

Cancel out x-20 in both numerator and denominator.

$$\frac{-2}{x+5}$$

Expand

$$-\frac{2}{x+5}$$

Short Solution Steps

Factor x^{2}-25. Factor x^{2}-10x+25.

$$\frac{\frac{x}{\left(x-5\right)\left(x+5\right)}-\frac{x-8}{\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

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

$$\frac{\frac{x\left(x-5\right)}{\left(x+5\right)\left(x-5\right)^{2}}-\frac{\left(x-8\right)\left(x+5\right)}{\left(x+5\right)\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

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

$$\frac{\frac{x\left(x-5\right)-\left(x-8\right)\left(x+5\right)}{\left(x+5\right)\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

Do the multiplications in x\left(x-5\right)-\left(x-8\right)\left(x+5\right).

$$\frac{\frac{x^{2}-5x-x^{2}-5x+8x+40}{\left(x+5\right)\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

Combine like terms in x^{2}-5x-x^{2}-5x+8x+40.

$$\frac{\frac{-2x+40}{\left(x+5\right)\left(x-5\right)^{2}}}{\frac{x-20}{\left(x-5\right)^{2}}}$$

Divide \frac{-2x+40}{\left(x+5\right)\left(x-5\right)^{2}} by \frac{x-20}{\left(x-5\right)^{2}} by multiplying \frac{-2x+40}{\left(x+5\right)\left(x-5\right)^{2}} by the reciprocal of \frac{x-20}{\left(x-5\right)^{2}}.

$$\frac{\left(-2x+40\right)\left(x-5\right)^{2}}{\left(x+5\right)\left(x-5\right)^{2}\left(x-20\right)}$$

Cancel out \left(x-5\right)^{2} in both numerator and denominator.

$$\frac{-2x+40}{\left(x-20\right)\left(x+5\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\left(-x+20\right)}{\left(x-20\right)\left(x+5\right)}$$

Extract the negative sign in 20-x.

$$\frac{-2\left(x-20\right)}{\left(x-20\right)\left(x+5\right)}$$

Cancel out x-20 in both numerator and denominator.

$$\frac{-2}{x+5}$$