ALG-19

Vieta sum (quadratic / cubic snapshot)

For ax² + bx + c, x1 + x2 = −b/a. For a cubic, x1+x2+x3 = −b/a.

Reading speed

PolynomialsVieta

Governing equation

x_1+x_2+\cdots=-b/a

$b \in \left[-20, 20\right]$

where

a: Leading a (—)
b: Next b (—)
\sum x_i: Sum of roots (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

x_1+x_2+\cdots=-b/a

$b \in \left[-20, 20\right]$

$b = -20$ $\sum x_i = 20$

Along the curve

②

Domain of the variable — blue

b_{\min} = -20,\quad b_{\max} = 20

$b \in \left[-20, 20\right]$

$b = -20$ $b = -20$

Along the curve

③

Response curve — green

x_1+x_2+\cdots=-b/a

$b \in \left[-20, 20\right]$

$b = -20$ $\sum x_i = 20$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sum x_i}{\mathrm{d}b}

$b \in \left[-20, 20\right]$

$b = -19.75$ $\sum x_i^{\prime} = -1$

Along the curve

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-19 — Vieta sum (quadratic / cubic snapshot)) is the form associated with Vieta. Working symbols: $a$ , $b$ $\rightarrow$ $\sum x_i$ . Vieta's formulae read elementary symmetric polynomials off the coefficients. Cardano is skipped; the sum survives.

Purpose

Purpose: compute

\sum x_i

from

a

b

in Algebra via

x_1+x_2+\cdots=-b/a

For ax² + bx + c, x1 + x2 = −b/a. For a cubic, x1+x2+x3 = −b/a. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 1.000\,\mathrm{—}

b = -6.000\,\mathrm{—}

, the governing relation

x_1+x_2+\cdots=-b/a

yields

\sum x_i = 6.0000\,\mathrm{—}

. Monic or not: enter a and b of x^n + … or ax^n + bx^{n−1} + … Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Leading a a1 —Next b b-6 —

Outputs

Sum of roots \sum x_i6.0000 —

Reading speed

Watch on YouTube

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Narration of this film

Monic or not: enter a and b of x^n + … or ax^n + bx^{n−1} + …

Vieta's formulae read elementary symmetric polynomials off the coefficients. Cardano is skipped; the sum survives.

Reading speed

Watch on YouTube