Sum of the Roots of a Quadratic

S = -\frac{b}{a}

Worked example: 2x² − 7x + 3 → root sum 3.5. Press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value.

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In this toolkit

The quadratic family8 formulas

Discriminant of a Quadratic: Both use coefficient of x² and quadratic coefficient b.Quadratic Formula (Negative Root): Both use coefficient of x² and quadratic coefficient b.Quadratic Formula (Positive Root): Both use coefficient of x² and quadratic coefficient b.Vertex x-Coordinate of a Parabola: Both use coefficient of x² and quadratic coefficient b.Vertex y-Coordinate of a Parabola: Both use coefficient of x² and quadratic coefficient b.Completing the Square: the Constant Needed: Both use quadratic coefficient b.

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Sum of the Roots of a Quadratic explained

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.

Sum of the Roots of a Quadratic formula

S = -\frac{b}{a}

Where

$S$ = Sum of the roots
$a$ = Coefficient of x²
$b$ = Coefficient of x

In this toolkit

The quadratic family

8 formulasThe two roots, the discriminant that predicts them, the vertex of the parabola and the sum-and-product shortcuts — all from ax² + bx + c.

Revised 31 October 2026

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formula.expert. (2026, October 31). Sum of the Roots of a Quadratic Formula & Calculator. https://formula.expert/solvers/quadratic-sum-of-roots

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