Quadratic Formula (Negative Root)

This is the minus branch of the quadratic formula — the same machine as the plus branch, with the square root subtracted instead of added. Because a solver hands back one number, the two roots live on two pages here; feed identical a, b and c to both and you have the full solution set. Worked example: 2x² + 3x − 5 = 0 has discriminant 9 + 40 = 49, so this branch gives x = (−3 − 7)/4 = −2.5 while the plus branch gives 1. Check it: 2(−2.5)² + 3(−2.5) − 5 = 12.5 − 7.5 − 5 = 0.

Two sanity checks catch most mistakes. The roots must sum to −b/a (here 1 + (−2.5) = −1.5 = −3/2 ✓) and multiply to c/a (1 × −2.5 = −2.5 = −5/2 ✓) — relations named for François Viète, who in 1591 became the first algebraist to use letters for the known coefficients as well as the unknowns, which is why we can write "ax² + bx + c" at all. When a is positive the minus branch is the smaller root; when a is negative the labels swap, because dividing by a negative 2a flips the order. And if the discriminant is exactly zero, both branches return the same repeated root.