Rechner zur Partialbruchzerlegung

Your input: perform the partial fraction decomposition of $$$\frac{1}{25 - x^{2}}$$$

Simplify the expression: $$$\frac{1}{25 - x^{2}}=\frac{-1}{x^{2} - 25}$$$

Factor the denominator: $$$\frac{-1}{x^{2} - 25}=\frac{-1}{\left(x - 5\right) \left(x + 5\right)}$$$

The form of the partial fraction decomposition is

$$\frac{-1}{\left(x - 5\right) \left(x + 5\right)}=\frac{A}{x - 5}+\frac{B}{x + 5}$$

Write the right-hand side as a single fraction:

$$\frac{-1}{\left(x - 5\right) \left(x + 5\right)}=\frac{\left(x - 5\right) B + \left(x + 5\right) A}{\left(x - 5\right) \left(x + 5\right)}$$

The denominators are equal, so we require the equality of the numerators:

$$-1=\left(x - 5\right) B + \left(x + 5\right) A$$

Expand the right-hand side:

$$-1=x A + x B + 5 A - 5 B$$

Collect up the like terms:

$$-1=x \left(A + B\right) + 5 A - 5 B$$

The coefficients near the like terms should be equal, so the following system is obtained:

$$\begin{cases} A + B = 0\\5 A - 5 B = -1 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=- \frac{1}{10}$$$, $$$B=\frac{1}{10}$$$

Therefore,

$$\frac{-1}{\left(x - 5\right) \left(x + 5\right)}=\frac{- \frac{1}{10}}{x - 5}+\frac{\frac{1}{10}}{x + 5}$$

Answer: $$$\frac{1}{25 - x^{2}}=\frac{- \frac{1}{10}}{x - 5}+\frac{\frac{1}{10}}{x + 5}$$$