Assume that for the quadratic equation $ax^2 + bx + c = 0$ there is no real solution ($\Delta = b^2 - 4ac < 0$). Is there a way to obtain an approximate real solution? i.e. some optimization technique.
Context - this is part of an algorithm I'm developing for a measurement system. Theoretically, the solution would always be real ($ \Delta \geq 0$); however, due to measurement errors this is not always the case. In most of these cases $\Delta$ is very close to zero.
Would assuming that $\Delta$ actually is equal to zero be the best possible approximation?
If you're just looking for the smallest absolute value of the quadratic, then you're looking for the vertex, which conveniently always occurs at $x = -\frac{b}{2a}$, with appropriate y-value.