How to do a quick estimation if $x_2 \ll x_1$ holds for the roots of a quadratic equation - to apply quick and easy root-finding formula? Wikipedia provides an interesting method of (approximately) solving a quadratic equation:

Vieta's formulas provide a useful method for finding the roots of a
quadratic in the case where one root is much smaller than the other.
If $|x_2|\ll|x_1|$, then $x_1+x_2\approx x_1$ and we have the estimate $$x_1\approx-\frac ba$$

But how can I quickly estimate that one root is much larger than the other?

One root is much smaller than the other when $|ac| \ll b^2$ because then the square root in the quadratic formula is very close to $b$.  The approximation given comes from replacing the square root by $b$ and taking the minus sign so the two terms add.  This is also the time that the calculation of the other root suffers from roundoff error in a computer because $-b+\sqrt{b^2-4ac}$ is the difference of two similar numbers.
