Partial Fraction Decomposition Calculator

Your input: perform the partial fraction decomposition of $$$\frac{x - 5}{x \left(x - 2\right)}$$$

The form of the partial fraction decomposition is

$$\frac{x - 5}{x \left(x - 2\right)}=\frac{A}{x}+\frac{B}{x - 2}$$

Write the right-hand side as a single fraction:

$$\frac{x - 5}{x \left(x - 2\right)}=\frac{x B + \left(x - 2\right) A}{x \left(x - 2\right)}$$

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

$$x - 5=x B + \left(x - 2\right) A$$

Expand the right-hand side:

$$x - 5=x A + x B - 2 A$$

Collect up the like terms:

$$x - 5=x \left(A + B\right) - 2 A$$

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

$$\begin{cases} A + B = 1\\- 2 A = -5 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=\frac{5}{2}$$$, $$$B=- \frac{3}{2}$$$

Therefore,

$$\frac{x - 5}{x \left(x - 2\right)}=\frac{\frac{5}{2}}{x}+\frac{- \frac{3}{2}}{x - 2}$$

Answer: $$$\frac{x - 5}{x \left(x - 2\right)}=\frac{\frac{5}{2}}{x}+\frac{- \frac{3}{2}}{x - 2}$$$