ALG-01

Quadratic formula

x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0.

Reading speed

AlgebraQuadratic formula

Governing equation

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

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

where

a: a (—)
b: b (—)
c: c (—)
x_+: Plus root (—)
x_-: Minus root (—)
\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

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

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

$b = -20$ $x_+ = 19.9$

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=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

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

$b = -20$ $x_+ = 19.9$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}x_+}{\mathrm{d}b}

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

$b = -19.75$ $x_+^{\prime} = -1.005$

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-01 — Quadratic formula) is the form associated with Quadratic formula. Working symbols: $a$ , $b$ , $c$ $\rightarrow$ $x_+$ , $x_-$ , $\Delta$ . Discriminant Δ = b² − 4ac: two reals, a double root, or a complex pair.

Purpose

Purpose: compute

x_+

x_-

\Delta

from

a

b

c

in Algebra via

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0. 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 = -3.000\,\mathrm{—}

c = 2.000\,\mathrm{—}

, the governing relation

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

yields

x_+ = 2.0000\,\mathrm{—}

x_- = 1.0000\,\mathrm{—}

\Delta = 1.0000\,\mathrm{—}

. Real branch: if Δ < 0 the sheet reports the real part as the repeated axis crossing of zero width. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a1 —b b-3 —c c2 —

Outputs

Plus root x_+2.0000 —
Minus root x_-1.0000 —
Discriminant \Delta1.0000 —

Reading speed

Watch on YouTube

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Narration of this film

Real branch: if Δ < 0 the sheet reports the real part as the repeated axis crossing of zero width.

Discriminant Δ = b² − 4ac: two reals, a double root, or a complex pair.

Reading speed

Watch on YouTube