Osamurtokehitelmän laskin

Your input: perform the partial fraction decomposition of $$$\frac{1}{5 - 3 x^{2}}$$$

Factor the denominator: $$$\frac{1}{5 - 3 x^{2}}=\frac{1}{- 3 \left(x - \frac{\sqrt{15}}{3}\right) \left(x + \frac{\sqrt{15}}{3}\right)}=- \frac{1}{3 x^{2} - 5}$$$

The form of the partial fraction decomposition is

$$\frac{- \frac{1}{3}}{\left(x - \frac{\sqrt{15}}{3}\right) \left(x + \frac{\sqrt{15}}{3}\right)}=\frac{A}{x - \frac{\sqrt{15}}{3}}+\frac{B}{x + \frac{\sqrt{15}}{3}}$$

Write the right-hand side as a single fraction:

$$\frac{- \frac{1}{3}}{\left(x - \frac{\sqrt{15}}{3}\right) \left(x + \frac{\sqrt{15}}{3}\right)}=\frac{3 \left(\left(3 x - \sqrt{15}\right) B + \left(3 x + \sqrt{15}\right) A\right)}{\left(3 x - \sqrt{15}\right) \left(3 x + \sqrt{15}\right)}$$

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

$$- \frac{1}{3}=3 \left(\left(3 x - \sqrt{15}\right) B + \left(3 x + \sqrt{15}\right) A\right)$$

Expand the right-hand side:

$$- \frac{1}{3}=x A + x B + \frac{\sqrt{15} A}{3} - \frac{\sqrt{15} B}{3}$$

Collect up the like terms:

$$- \frac{1}{3}=x \left(A + B\right) + \frac{\sqrt{15} A}{3} - \frac{\sqrt{15} B}{3}$$

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

$$\begin{cases} A + B = 0\\\frac{\sqrt{15} A}{3} - \frac{\sqrt{15} B}{3} = - \frac{1}{3} \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=- \frac{\sqrt{15}}{30}$$$, $$$B=\frac{\sqrt{15}}{30}$$$

Therefore,

$$\frac{- \frac{1}{3}}{\left(x - \frac{\sqrt{15}}{3}\right) \left(x + \frac{\sqrt{15}}{3}\right)}=\frac{- \frac{\sqrt{15}}{10}}{3 x - \sqrt{15}}+\frac{\frac{\sqrt{15}}{10}}{3 x + \sqrt{15}}$$

Answer: $$$\frac{1}{5 - 3 x^{2}}=\frac{- \frac{\sqrt{15}}{10}}{3 x - \sqrt{15}}+\frac{\frac{\sqrt{15}}{10}}{3 x + \sqrt{15}}$$$