## Task 2 - 050812

For which real numbers $a$ and $b$ is the equation

$$
\frac{1}{a}+\frac{1}{b}=\frac{(a+b) \cdot(a-b)}{a b} \quad \text{ satisfied? }
$$
Neither $a$ nor $b$ may be zero. Then we get

$$
\frac{a+b}{a b}=\frac{(a+b) \cdot(a-b)}{a b} \Rightarrow 0=(a+b) \cdot(a-b)-(a+b) \Rightarrow(a+b)(a-b-1)=0
$$

A product is zero if at least one factor is zero. Therefore, $a+b=0$ or $a-b=1$ must hold. Conversely, it is obvious that the equation is satisfied if either $a=-b$ or $a=b+1$ holds.