Let's consider $$x^2 + 2px - q = 0$$ We know the roots to be $$x = -p \pm \sqrt{p^2 +q}$$ So let's take a look at $$x = -p + \sqrt{p^2 + q}$$

Let's take $p$ very large and $q$ to be small:

In [8]:
from math import sqrt
p = 1e6
q = 0.1

x = -p + sqrt(p**2 + q)
print(repr(x))
print(repr(x**2 + 2*p*x - q))

4.9942173063755035e-08
-0.00011565387248743675


Is this accurate? Not quite. Let's try rearranging: $$\frac{q}{p + \sqrt{p^2 + q}}$$

In [9]:
x = q / (p + sqrt(p**2 + q))
print(repr(x))
print(repr(x**2 + 2*p*x - q))

4.999999999999876e-08
1.3877787807814457e-17

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