Factorise $18 861 649$. Hence or otherwise solve the quadratic
$$ x^2 − 18 861 649 = 0$$
We find that $\sqrt{18861649} = 4343$ exactly. So we can work on 4343.
$\lceil \sqrt{4343} \rceil = 66$, and $66^2 - 4343 = 13$ which is not a perfect square.
By additional trials we find $72^2 - 4343 = 29^2$, and so $4343 = (72-29)(72+29) = 43 \times 101$.
So the factorisation is $18861649 = 43^2 \times 101^2$.
Using the above factorisation we have
$$ \begin{align} 0 & = x^2 - 18861649 \\ \\ &= x^2 - (43 \times 101)^2 \\ \\ & = (x - 4343)(x+4343) \end{align} $$
And so $x = \pm 4343$.