ALG-07

Completing-the-square snapshot

ax² + bx + c = a(x + b/2a)² + (c − b²/4a). Vertex form.

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AlgebraCompleting the square

Governing equation

ax^2+bx+c=a\left(x+\dfrac{b}{2a}\right)^2+k

where

a: a (—)
b: b (—)
c: c (—)
h: Vertex x (—)
k: Vertex y (—)

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-07 — Completing-the-square snapshot) is the form associated with Completing the square. Working symbols: $a$ , $b$ , $c$ $\rightarrow$ $h$ , $k$ . The shift x ↦ x + b/(2a) kills the linear term. The vertex is at x = −b/(2a).

Purpose

Purpose: compute

h

k

from

a

b

c

in Algebra via

ax^2+bx+c=a\left(x+\dfrac{b}{2a}\right)^2+k

ax² + bx + c = a(x + b/2a)² + (c − b²/4a). Vertex form. 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{—}

c = 5.000\,\mathrm{—}

, the governing relation

ax^2+bx+c=a\left(x+\dfrac{b}{2a}\right)^2+k

yields

h = 3.0000\,\mathrm{—}

k = -4.0000\,\mathrm{—}

. Real quadratic, output vertex and the constant remainder k. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a1 —b b-6 —c c5 —

Outputs

Vertex x h3.0000 —
Vertex y k-4.0000 —

Reading speed

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Narration of this film

Real quadratic, output vertex and the constant remainder k.

The shift x ↦ x + b/(2a) kills the linear term. The vertex is at x = −b/(2a).

Reading speed

Watch on YouTube