Sum of the Roots of a Quadratic

You can read the sum of a quadratic's two roots straight off its coefficients: expand a(x − x₁)(x − x₂) and the x term comes out as −a(x₁ + x₂), so x₁ + x₂ = −b/a. Worked example: 2x² − 7x + 3 = 0 has S = 7/2 = 3.5, and the actual roots 3 and ½ do indeed add to 3.5 — no formula, no square roots, no arithmetic beyond one division.

The relation carries François Viète's name; his 1591 In artem analyticem isagoge introduced the habit of writing consonants for known coefficients and vowels for unknowns, and only with symbolic coefficients could such a statement even be phrased. Today it is the fastest way to check a solution — if your two roots do not sum to −b/a you have made an arithmetic slip — and the fastest way to build a quadratic to order: roots summing to 5 and multiplying to 6 give x² − 5x + 6 = 0. Two traps: the minus sign is part of the formula (S = −b/a, not b/a), and it holds even when the roots are complex, where the imaginary parts cancel and the sum stays real.