I was wondering, if when solving a quadratic equation where factoring wasn't helpful in finding a solution, if there were any cases in which using the quadratic formula would generate results that would not solve the equation? My understanding is that the quadratic formula tends to be a shortcut in using the method of 'completing the square' and I wondered if there were any way in which to discern when using these methods that they would not resolve in a constructive solution, other than those involving imaginary numbers(multiples of the square root of -1)?
2026-04-07 22:56:04.1775602564
Quadratic Formula: Any Shortcomings?
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In doing numerical computations, the quadratic formula can in some cases produce "catastrophic cancellation" where you subtract two numbers that are very close together, resulting in a large relative error. For example, consider
$$ 10^{-20} x^2 + x + 1 = 0 $$
Your calculator is likely to get $\sqrt{b^2 - 4 a c} = 1.0$, resulting in a root of $0$. But in fact the root near $0$ is approximately $-1$.