33.2 Roots of Quadratic Polynomials

Here we solve

$$x^2+a_1x+a_0 = 0,$$ (33.1)

where $a_1$ and $a_0$ are real. Well known techniques (larger magnitude root $x_1$ computed by quadratic formula, smaller magnitude root computed by applying Vieta's rule: $x_2=a_0/x_1$) are used to ensure numerical accuracy of the roots. Additionally the code is equipped to deal with extremely large coefficients, such that computational overflow will not occur. A coefficient rescaling technique is applied if during evaluation of the quadratic formula either the term $(a_1/2)^2$ or the discriminant $(a_1/2)^2-a_0$ become larger than the largest positive number on the machine.