ALG-08

Binomial (a+b)²

(a+b)² = a² + 2ab + b².

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AlgebraBinomial

Governing equation

(a+b)^2=a^2+2ab+b^2

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

where

a: a (—)
b: b (—)
(a+b)^2: Square (—)
2ab: Cross term (—)

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

(a+b)^2=a^2+2ab+b^2

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

$b = -20$ $(a+b)^2 = 289$

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

(a+b)^2=a^2+2ab+b^2

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

$b = -20$ $(a+b)^2 = 289$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}(a+b)^2}{\mathrm{d}b}

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

$b = -19.75$ $(a+b)^2^{\prime} = -33.5$

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-08 — Binomial (a+b)²) is the form associated with Binomial. Working symbols: $a$ , $b$ $\rightarrow$ $(a+b)^2$ , $2ab$ . The square of a sum. Also (a−b)² = a² − 2ab + b² and a² − b² = (a−b)(a+b).

Purpose

Purpose: compute

(a+b)^2

2ab

from

a

b

in Algebra via

(a+b)^2=a^2+2ab+b^2

(a+b)² = a² + 2ab + b². 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 = 3.000\,\mathrm{—}

b = 4.000\,\mathrm{—}

, the governing relation

(a+b)^2=a^2+2ab+b^2

yields

(a+b)^2 = 49.0000\,\mathrm{—}

2ab = 24.0000\,\mathrm{—}

. Two reals, expanded square and the cross term. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a3 —b b4 —

Outputs

Square (a+b)^249.0000 —
Cross term 2ab24.0000 —

Reading speed

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Narration of this film

Two reals, expanded square and the cross term.

The square of a sum. Also (a−b)² = a² − 2ab + b² and a² − b² = (a−b)(a+b).

Reading speed

Watch on YouTube