Sum and Product of Roots: Vieta's Formulas with Examples

If ax² + bx + c = 0 has roots α and β, then α + β = −b/a and αβ = c/a. These two relations are called Vieta's formulas, named after the French mathematician François Viète, who discovered how the roots of a polynomial connect directly to its coefficients without any solving.

They matter because they let you work with the roots of a quadratic without actually finding them. You can check whether a guessed pair of roots is correct, build new equations from given roots, or evaluate symmetric expressions like α² + β² = (α + β)² − 2αβ in seconds.

Key Concepts

• Root: a value of x that makes the polynomial equal zero. A quadratic has exactly two roots when counted with multiplicity; they may be equal or complex.
• Coefficient: the numbers a, b, c in ax² + bx + c = 0, with the strict requirement a ≠ 0.
• Vieta's formulas (quadratic): sum of roots α + β = −b/a and product of roots αβ = c/a.
• Complex roots included: the formulas stay true even when the discriminant b² − 4ac is negative and the roots are complex numbers.
• Cubic extension: for ax³ + bx² + cx + d = 0 with roots α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, and αβγ = −d/a.

Formulas

Write the quadratic in factored form: a(x − α)(x − β) = ax² − a(α + β)x + aαβ. Comparing coefficients with ax² + bx + c gives b = −a(α + β) and c = aαβ, which rearrange to the formulas below. Assumptions: a ≠ 0, and the formulas hold for real and complex roots alike.

• α + β = −b/a (sum of roots)
• αβ = c/a (product of roots)
• α² + β² = (α + β)² − 2αβ (derived symmetric expression)
• 1/α + 1/β = (α + β)/αβ, valid only when αβ ≠ 0

Worked Example

Find the sum and product of the roots of 2x² − 5x − 3 = 0, then verify with the actual roots. Here a = 2, b = −5, c = −3. Sum = −b/a = −(−5)/2 = 5/2. Product = c/a = −3/2. Now find the roots with the quadratic formula: x = [5 ± √(25 + 24)]/4 = [5 ± 7]/4, giving x = 3 and x = −1/2. Check the sum: 3 + (−1/2) = 5/2, which matches. Check the product: 3 × (−1/2) = −3/2, which also matches. Both formulas are confirmed.

Common Misconceptions

• "The sum of the roots is b/a." The sign is wrong: it is −b/a. For x² − 7x + 12 = 0 the sum is 7, not −7.
• "Vieta's formulas need real roots." They hold for complex roots too; nothing in the derivation assumes the discriminant is non-negative.
• "They work for any equation." They apply to polynomials only, and a must not be zero — if a = 0 the equation is linear, not quadratic, and the formulas are meaningless.

Real-World Applications

• Checking factorisations quickly in algebra problems and exams.
• Forming quadratic equations from given roots (the reverse process).
• Analysing parabolas: the vertex sits at x = −b/2a, exactly midway between the two roots.
• Physics problems with quadratic models, such as projectile motion, where root relations give flight times without solving.

Vieta's formulas link a quadratic's roots to its coefficients through α + β = −b/a and αβ = c/a. They work for real and complex roots and extend naturally to cubics.