Rechner zur Partialbruchzerlegung

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

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

The form of the partial fraction decomposition is

$$\frac{-1}{\left(x - \sqrt{2}\right) \left(x + \sqrt{2}\right)}=\frac{A}{x + \sqrt{2}}+\frac{B}{x - \sqrt{2}}$$

Write the right-hand side as a single fraction:

$$\frac{-1}{\left(x - \sqrt{2}\right) \left(x + \sqrt{2}\right)}=\frac{\left(x - \sqrt{2}\right) A + \left(x + \sqrt{2}\right) B}{\left(x - \sqrt{2}\right) \left(x + \sqrt{2}\right)}$$

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

$$-1=\left(x - \sqrt{2}\right) A + \left(x + \sqrt{2}\right) B$$

Expand the right-hand side:

$$-1=x A + x B - \sqrt{2} A + \sqrt{2} B$$

Collect up the like terms:

$$-1=x \left(A + B\right) - \sqrt{2} A + \sqrt{2} B$$

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

$$\begin{cases} A + B = 0\\- \sqrt{2} A + \sqrt{2} B = -1 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=\frac{\sqrt{2}}{4}$$$, $$$B=- \frac{\sqrt{2}}{4}$$$

Therefore,

$$\frac{-1}{\left(x - \sqrt{2}\right) \left(x + \sqrt{2}\right)}=\frac{\frac{\sqrt{2}}{4}}{x + \sqrt{2}}+\frac{- \frac{\sqrt{2}}{4}}{x - \sqrt{2}}$$

Answer: $$$\frac{1}{2 - x^{2}}=\frac{\frac{\sqrt{2}}{4}}{x + \sqrt{2}}+\frac{- \frac{\sqrt{2}}{4}}{x - \sqrt{2}}$$$