Partial Fraction Decomposition Calculator

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

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

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

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

Collect up the like terms:

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

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,

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

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