Task B-1.3. If $a+b=2$ and $a^{2}+b^{2}=6$, what is $a^{-1}+b^{-1}$?
Solution.

$$
a^{-1}+b^{-1}=\frac{1}{a}+\frac{1}{b}=\frac{b+a}{a b}=\frac{2}{a b}
$$

Let's square the expression $a+b=2$. From

$$
a^{2}+2 a b+b^{2}=4
$$

it follows that $2 a b=4-6$, so $a b=-1$.

Then we have

$$
a^{-1}+b^{-1}=-2
$$