Roots Relationships (Vieta's Formulas)

Explore the elegant connection between a polynomial's roots and its coefficients. For a monic quadratic $x^2 + bx + c = 0$, the sum of roots equals $-b$, and their product equals $c$.

Interactive Sandbox

Roots & Vieta Geometry Plotter

Drag the roots directly on the x-axis or use the sliders. Observe how the roots construct the parabola, and visualize the Sum ($r_1 + r_2$) as a 1D vector and the Product ($r_1 \cdot r_2$) as a 2D area.

Quadratic Plot: y = (x - r₁)(x - r₂)
Vieta Geometry: Sum (Vector) & Product (Area)

Active Function

y = x² - 0.00x + 0.00
Root 1 (r₁) -2.00
Root 2 (r₂) 3.00
Drag the cyan and amber dots on the x-axis of the top plot to adjust the roots interactively!
Arithmetic Solver

Step-by-Step Algebraic Expansion

Trace how factoring and expanding polynomials mathematically derives the relationship between roots and coefficients.

Deriving Coefficients from Roots


1. Linear Factors:
f(x) = (x - r₁)(x - r₂)
f(x) = (x - (-2.00))(x - (3.00))

2. FOIL Expansion:
f(x) = x² - r₂x - r₁x + r₁r₂
f(x) = x² - (3.00)x - (-2.00)x + (-2.00 × 3.00)
f(x) = x² - (r₁ + r₂)x + (r₁ × r₂)

3. Evaluate Vieta's Coefficients:
Sum (S) = r₁ + r₂ = 1.00  &implies;  Coefficient b = -S = -1.00
Product (P) = r₁ × r₂ = -6.00  &implies;  Coefficient c = P = -6.00
f(x) = x² - 1.00x - 6.00

Vieta Coefficients Summary


Linear Coefficient (b)

b = -(r₁ + r₂) = -1.00

Constant Term (c)

c = r₁ × r₂ = -6.00

Evaluation

Conceptual Quiz

Test your understanding of root relationships, Vieta's formulas, and coefficient derivation.

Question 1 of 5 Score: 0/0

For the quadratic equation x² - 5x + 6 = 0, what is the sum and product of the roots?

Correct Answer!

Explanation text goes here...

Vieta's Rules Cheat Sheet


  • Monic: x² + bx + c = (x - r₁)(x - r₂)
  • Sum: r₁ + r₂ = -b
  • Product: r₁ × r₂ = c
  • General: ax² + bx + c = 0 &implies; r₁ + r₂ = -b/a, r₁r₂ = c/a