Calcolatore di decomposizione in fratti semplici

Your input: perform the partial fraction decomposition of $$$\frac{1}{x^{2} - 78 x}$$$

Simplify the expression: $$$\frac{1}{x^{2} - 78 x}=\frac{1}{x \left(x - 78\right)}$$$

The form of the partial fraction decomposition is

$$\frac{1}{x \left(x - 78\right)}=\frac{A}{x}+\frac{B}{x - 78}$$

Write the right-hand side as a single fraction:

$$\frac{1}{x \left(x - 78\right)}=\frac{x B + \left(x - 78\right) A}{x \left(x - 78\right)}$$

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

$$1=x B + \left(x - 78\right) A$$

Expand the right-hand side:

$$1=x A + x B - 78 A$$

Collect up the like terms:

$$1=x \left(A + B\right) - 78 A$$

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

$$\begin{cases} A + B = 0\\- 78 A = 1 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=- \frac{1}{78}$$$, $$$B=\frac{1}{78}$$$

Therefore,

$$\frac{1}{x \left(x - 78\right)}=\frac{- \frac{1}{78}}{x}+\frac{\frac{1}{78}}{x - 78}$$

Answer: $$$\frac{1}{x^{2} - 78 x}=\frac{- \frac{1}{78}}{x}+\frac{\frac{1}{78}}{x - 78}$$$