Decompose x/(x - 2)^2 into partial fractions.

Step-by-step solution

1. 1

   Rewrite the numerator in terms of the denominator's base. The denominator is built from $x-2$, so express $x$ using that same block:

   $x=(x-2)+2$

   This is an identity, so nothing has been assumed — it simply reorganises the numerator.

2. 2

   Substitute and split the single fraction into two.

   $\frac{x}{(x-2)^2}=\frac{(x-2)+2}{(x-2)^2}=\frac{x-2}{(x-2)^2}+\frac{2}{(x-2)^2}$

3. 3

   Cancel in the first fraction. One factor of $(x-2)$ divides out:

   $\frac{x-2}{(x-2)^2}=\frac{1}{x-2}$

   $\frac{x}{(x-2)^2}=\frac{1}{x-2}+\frac{2}{(x-2)^2},\qquad x\neq 2$

4. 4

   Compare with the algebraic method. Setting $\tfrac{x}{(x-2)^2}=\tfrac{A}{x-2}+\tfrac{B}{(x-2)^2}$ gives $x=A(x-2)+B$, so $A=1$ and $B=2$ — the same answer. The numerator trick reaches it in one line and is worth recognising whenever the numerator is linear and the denominator is a repeated linear factor.

5. 5

   Check numerically. At $x=4$: left $=\tfrac44=1$, right $=\tfrac12+\tfrac24=1$ $\checkmark$. At $x=0$: left $=0$, right $=-\tfrac12+\tfrac24=0$ $\checkmark$.