Suppose a machine with the floating-point system $\beta = 10$, $p = 8$, $L = -50$, and $U = 50$ is used to calculate the roots of a quadratic equation $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are given real coefficients. For each of the following, state the numerical difficulties that arise when using the standard formula for computing the root. Explain how to overcome these difficulties when possible.
I think the second one and third one have cancellation error since $b^2 \gg 4ac$, However, the first one is $b^2 \ll 4ac$, can someone what kind of error it causes and how to fix it.
When $b^2 \ll 4ac$, there is not much of a problem. You can lose the $b^2$ term compared with $4ac$ when the ratio is smaller than the machine $\epsilon$, but that is reality. In this case you do need complex numbers, but that, too, is reality.