About

Vieta's formulas provide a direct relationship between the roots of a polynomial and its coefficients. This tool focuses on Quadratic Equations (ax² + bx + c = 0). Instead of just solving for x, it demonstrates the algebraic link: the sum of the roots equals -b/a and the product equals c/a.

This utility features a dual-mode interface. The "Coefficients to Roots" mode verifies sums and products, often used to check mental math or homework. The "Roots to Equation" mode is a reverse engineering tool: input two roots (real or complex), and the calculator reconstructs the original integer coefficients of the quadratic equation.

Formulas

Quadratic Equation:

ax² + bx + c = 0

Vieta's Relations:

{

x₁ + x₂ = −bax₁ ⋅ x₂ = ca

Reference Data

Relation	Formula	Description
Sum of Roots	x₁ + x₂ = −b/a	Negated linear coeff over quadratic coeff
Product of Roots	x₁ ⋅ x₂ = c/a	Constant term over quadratic coeff
Quadratic Form	ax² + bx + c = 0	Standard Form
Factored Form	a(x − x₁)(x − x₂) = 0	Using roots directly
Discriminant	Δ = b² − 4ac	Determines root type
Root Type	Δ > 0	2 Distinct Real Roots
Root Type	Δ = 0	1 Repeated Real Root
Root Type	Δ < 0	2 Complex Conjugate Roots

Frequently Asked Questions

If b^2 - 4ac < 0, the equation has no real roots. However, Vieta's formulas still hold true for complex numbers (imaginary roots involving i). The sum of complex conjugate roots will always be a real number.

Vieta's formulas generalize to any polynomial degree. For a cubic equation (ax^3 + bx^2 + cx + d), the sum of roots is still -b/a, but the product is -d/a. This specific calculator is optimized for quadratics.

If "a" is zero, the term ax^2 vanishes, and the equation becomes linear (bx + c = 0). Linear equations have only one root, so the concepts of sum and product of two roots do not apply in the standard quadratic sense.

Yes. The formulas work for any real numbers, including fractions and decimals. The calculator will process decimal inputs accurately.