ALG-18

Quadratic discriminant

Δ = b² − 4ac. Sign of Δ classifies the roots.

Reading speed

AlgebraDiscriminant

Governing equation

\Delta=b^2-4ac

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

where

a: a (—)
b: b (—)
c: c (—)
\Delta: Discriminant (—)

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

\Delta=b^2-4ac

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

$b = -20$ $\Delta = 384$

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

\Delta=b^2-4ac

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

$b = -20$ $\Delta = 384$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta}{\mathrm{d}b}

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

$b = -19.75$ $\Delta^{\prime} = -39.5$

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-18 — Quadratic discriminant) is the form associated with Discriminant. Working symbols: $a$ , $b$ , $c$ $\rightarrow$ $\Delta$ . Δ > 0 two reals, Δ = 0 a double root, Δ < 0 a complex conjugate pair.

Purpose

Purpose: compute

\Delta

from

a

b

c

in Algebra via

\Delta=b^2-4ac

Δ = b² − 4ac. Sign of Δ classifies the roots. 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 = 5.000\,\mathrm{—}

c = 4.000\,\mathrm{—}

, the governing relation

\Delta=b^2-4ac

yields

\Delta = 9.0000\,\mathrm{—}

. Coefficients of ax² + bx + c. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a1 —b b5 —c c4 —

Outputs

Discriminant \Delta9.0000 —

Reading speed

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Narration of this film

Coefficients of ax² + bx + c.

Δ > 0 two reals, Δ = 0 a double root, Δ < 0 a complex conjugate pair.

Reading speed

Watch on YouTube