A careful floating-point computer implementation combines several strategies to produce a robust result. Assuming that the discriminant b² − 4ac is positive, and b is nonzero, the computation would be as follows:^[1]

${\displaystyle {\begin{aligned}x_{1}&={\frac {-b-\operatorname {sgn}(b)\,{\sqrt {b^{2}-4ac}}}{2a}},\\x_{2}&={\frac {2c}{-b-\operatorname {sgn}(b)\,{\sqrt {b^{2}-4ac}}}}={\frac {c}{ax_{1}}}.\end{aligned}}}$

Here sgn denotes the sign function, where ${\displaystyle \operatorname {sgn}(b)}$ is 1 if ${\displaystyle b}$ is positive, and −1 if ${\displaystyle b}$ is negative. This avoids cancellation problems between ${\displaystyle b}$ and the square root of the discriminant by ensuring that only numbers of the same sign are added.

To illustrate the instability of the standard quadratic formula compared to this formula, consider a quadratic equation with roots ${\displaystyle 1.786737589984535}$ and ${\displaystyle 1.149782767465722\times 10^{-8}}$. To 16 significant digits, roughly corresponding to double-precision accuracy on a computer, the monic quadratic equation with these roots may be written as

${\displaystyle x^{2}-1.786737601482363x+2.054360090947453\times 10^{-8}=0.}$

Using the standard quadratic formula and maintaining 16 significant digits at each step, the standard quadratic formula yields

${\displaystyle {\sqrt {\Delta }}=1.786737578486707,}$

${\displaystyle x_{1}=(1.786737601482363+1.786737578486707)/2=1.786737589984535,}$

${\displaystyle x_{2}=(1.786737601482363-1.786737578486707)/2=0.000000011497828.}$

Note how cancellation has resulted in ${\displaystyle x_{2}}$ being computed to only 8 significant digits of accuracy.

The variant formula presented here, however, yields the following:

${\displaystyle x_{1}=(1.786737601482363+1.786737578486707)/2=1.786737589984535,}$

${\displaystyle x_{2}=2.054360090947453\times 10^{-8}/1.786737589984535=1.149782767465722\times 10^{-8}.}$

Note the retention of all significant digits for ${\displaystyle x_{2}}$.

Note that while the above formulation avoids catastrophic cancellation between ${\displaystyle b}$ and ${\displaystyle {\sqrt {b^{2}-4ac}}}$, there remains a form of cancellation between the terms ${\displaystyle b^{2}}$ and ${\displaystyle -4ac}$ of the discriminant, which can still lead to loss of up to half of correct significant digits.^[2][3] The discriminant ${\displaystyle b^{2}-4ac}$ needs to be computed in arithmetic of twice the precision of the result to avoid this (e.g. quad precision if the final result is to be accurate to full double precision).^[4] This can be in the form of a fused multiply-add operation.^[2]

To illustrate this, consider the following quadratic equation, adapted from Kahan (2004):^[2]

${\displaystyle 94906265.625x^{2}-189812534x+94906268.375.}$

This equation has ${\displaystyle \Delta =7.5625}$ and roots

${\displaystyle x_{1}=1.000000028975958,}$

${\displaystyle x_{2}=1.000000000000000.}$

However, when computed using IEEE 754 double-precision arithmetic corresponding to 15 to 17 significant digits of accuracy, ${\displaystyle \Delta }$ is rounded to 0.0, and the computed roots are

${\displaystyle x_{1}=1.000000014487979,}$

${\displaystyle x_{2}=1.000000014487979,}$

which are both false after the 8th significant digit. This is despite the fact that superficially, the problem seems to require only 11 significant digits of accuracy for its solution.