Solve x/x^2-25-4/x^2-10x+25

$$\frac { x } { x ^ { 2 } - 25 } - \frac { 4 } { x ^ { 2 } - 10 x + 25 }$$

Evaluate

$$\frac{x^{2}-9x-20}{\left(x+5\right)\left(x-5\right)^{2}}$$

Short Solution Steps

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

$$\frac{x}{\left(x-5\right)\left(x+5\right)}-\frac{4}{\left(x-5\right)^{2}}$$

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{4}{\left(x-5\right)^{2}} times \frac{x+5}{x+5}.

$$\frac{x\left(x-5\right)}{\left(x+5\right)\left(x-5\right)^{2}}-\frac{4\left(x+5\right)}{\left(x+5\right)\left(x-5\right)^{2}}$$

Since \frac{x\left(x-5\right)}{\left(x+5\right)\left(x-5\right)^{2}} and \frac{4\left(x+5\right)}{\left(x+5\right)\left(x-5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x-5\right)-4\left(x+5\right)}{\left(x+5\right)\left(x-5\right)^{2}}$$

Do the multiplications in x\left(x-5\right)-4\left(x+5\right).

$$\frac{x^{2}-5x-4x-20}{\left(x+5\right)\left(x-5\right)^{2}}$$

Combine like terms in x^{2}-5x-4x-20.

$$\frac{x^{2}-9x-20}{\left(x+5\right)\left(x-5\right)^{2}}$$

Expand \left(x+5\right)\left(x-5\right)^{2}.

$$\frac{x^{2}-9x-20}{x^{3}-5x^{2}-25x+125}$$

Differentiate w.r.t. x

$$\frac{325+55x+13x^{2}-x^{3}}{\left(x+5\right)^{2}\left(x-5\right)^{3}}$$