Completing the Square — Mathematics Knowledge System

Overview

Completing the square is a technique for rewriting a quadratic expression ax² + bx + c as a perfect square trinomial plus (or minus) a constant, a(x - h)² + k. This single algebraic maneuver unlocks the vertex of a parabola directly, gives the cleanest possible derivation of the quadratic formula, and is the essential first step for converting circle, ellipse, and hyperbola equations into standard form. It converts a quadratic that's awkward to reason about into one whose maximum/minimum value and axis of symmetry are immediately visible.

Intuition

Picture x² + bx as an actual square of side x with a rectangle of width b attached to one side, total area x² + bx. This L-shaped figure isn't a square yet — it's missing a corner. If you split the rectangle in half (each half has width b/2) and fold one half around to the adjacent side, you almost form a bigger square of side (x + b/2) — except there's a small square-shaped hole in the corner, with area (b/2)². Fill that hole in, and you've completed the square: x² + bx + (b/2)² = (x + b/2)². This geometric picture is exactly why the algebraic recipe is 'take half the coefficient of x, square it, and add it.'

Formal Definition

Definition

For a quadratic ax² + bx + c, completing the square rewrites it in vertex form:

ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 + \left(c-\frac{b^2}{4a}\right)

General identity

a(x-h)^2+k, \quad h=-\frac{b}{2a},\ k=c-\frac{b^2}{4a}

(h,k) is the vertex of the parabola y = ax² + bx + c

Vertex form

Notation

Notation	Meaning
$a(x-h)^2+k$	Vertex form of a quadratic; (h,k) is the vertex
$\left(\frac{b}{2}\right)^2$	The 'completing' constant added to x² + bx to form a perfect square trinomial

Derivation

Completing the square for the monic case x² + bx + c step by step:

x^2 + bx + c

Start with the expression

= \left(x^2 + bx + \left(\tfrac{b}{2}\right)^2\right) - \left(\tfrac{b}{2}\right)^2 + c

Add and immediately subtract (b/2)² so the value of the expression is unchanged

= \left(x+\tfrac{b}{2}\right)^2 - \tfrac{b^2}{4} + c

The first three terms form a perfect square trinomial: x² + bx + (b/2)² = (x + b/2)²

= \left(x+\tfrac{b}{2}\right)^2 + \left(c - \tfrac{b^2}{4}\right)

Combine the constants — this is vertex form with h = -b/2, k = c - b²/4

Proofs

x² + bx + (b/2)² is a perfect square trinomial

$\left(x+\tfrac{b}{2}\right)^2 = \left(x+\tfrac{b}{2}\right)\left(x+\tfrac{b}{2}\right)$ (Definition of squaring)
$= x^2 + \tfrac{b}{2}x + \tfrac{b}{2}x + \tfrac{b^2}{4}$ (FOIL / distribute)
$= x^2 + bx + \tfrac{b^2}{4}$ (Combine like terms: (b/2)x + (b/2)x = bx)
$\therefore x^2+bx+\left(\tfrac{b}{2}\right)^2 = \left(x+\tfrac{b}{2}\right)^2$ (This confirms the reverse direction used when completing the square)

Properties

Vertex identification

\text{The vertex of } y=ax^2+bx+c \text{ is } \left(-\tfrac{b}{2a}, c-\tfrac{b^2}{4a}\right)

Leading coefficient factor first

\text{For } a \neq 1, \text{ factor } a \text{ out of the } x^2 \text{ and } x \text{ terms before completing the square}

Sign of a determines max/min

a>0 \Rightarrow k \text{ is a minimum}; \quad a<0 \Rightarrow k \text{ is a maximum}

Route to the quadratic formula

\text{Completing the square on } ax^2+bx+c=0 \text{ derives } x = \tfrac{-b\pm\sqrt{b^2-4ac}}{2a}

Theorems

Theorem 1: Vertex Form Theorem

\text{Every quadratic } f(x) = ax^2+bx+c \ (a\neq 0) \text{ can be written uniquely as } f(x)=a(x-h)^2+k \text{ where } h=-b/2a, \ k = f(h).

Applications

Completing the square on projectile motion equations h(t) = -½gt² + v₀t + h₀ reveals the maximum height and the time it occurs, without calculus.

Formula Explorer

Animation

Animates the classic geometric proof: a square of side x with an attached b-wide rectangle splits into two b/2-wide rectangles, one of which rotates to the adjacent side, leaving a (b/2)×(b/2) gap that is then filled in to complete a larger square — visually deriving the 'take half, square it' rule.

Worked Examples

Move the constant to the other side.
$x^2+8x = -10$
Take half of 8 (=4), square it (=16), and add to both sides.
$x^2+8x+16 = -10+16$
Write the left side as a perfect square and simplify the right.
$(x+4)^2 = 6$
Take the square root of both sides and solve.
$x+4 = \pm\sqrt{6} \;\Rightarrow\; x = -4\pm\sqrt{6}$

Answer: x = -4 ± √6 (vertex form: (x+4)² - 6 = 0)

Practice Problems

Difficulty 3/10

Complete the square to solve x² - 10x + 21 = 0.

Difficulty 4/10

Write g(x) = -x² + 4x + 1 in vertex form and state the maximum value.

Difficulty 5/10

A ball's height is h(t) = -16t² + 96t + 4 (feet). Use completing the square to find the maximum height and when it occurs.

Common Mistakes

Common Mistake

Adding (b/2)² to only one side of the equation instead of both.

When solving an equation, whatever constant you add to complete the square on the left must also be added to the right side to preserve equality.

Common Mistake

Forgetting to factor the leading coefficient a out of the x² and x terms before completing the square when a ≠ 1.

The 'take half, square it' shortcut only works on an expression of the form x² + bx (leading coefficient exactly 1). If a ≠ 1, factor it out of the variable terms first, complete the square inside the parentheses, then distribute a back.

Common Mistake

Sign errors when writing the vertex form, e.g. writing (x + h)² instead of (x - h)² for a positive h.

Vertex form is a(x - h)² + k. If completing the square produces (x + 3)², that means h = -3, so the vertex's x-coordinate is -3, not +3 — always double check by expanding.

Quiz

What constant completes the square for x² + 12x?

12636144

The vertex form of a quadratic is a(x-h)² + k. What does the sign of a tell you?

Whether the parabola is wider or narrowerWhether (h,k) is a maximum or a minimumThe x-intercepts of the parabolaNothing about the vertex

Flashcards

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Historical Background

Completing the square predates modern algebraic notation by millennia — it was originally a literal geometric procedure. Babylonian mathematicians (c. 1800–1600 BCE) solved quadratic problems using cut-and-rearrange geometric methods equivalent to completing the square, recorded on clay tablets, though without symbolic algebra. The Persian mathematician al-Khwarizmi, in his 9th-century treatise on al-jabr, gave an explicit geometric proof: he literally drew a square of side x, attached rectangles representing the bx term, and then added a small corner square to 'complete' the larger square — the same picture used in classrooms today. The technique remained a geometric argument until Renaissance and early modern algebraists (Viète, Descartes) translated it into pure symbolic manipulation, at which point it became the standard route to deriving the quadratic formula.

c. 1800–1600 BCE
Babylonian scribes solve quadratic problems using geometric cut-and-paste methods equivalent to completing the square
c. 820 CE
Al-Khwarizmi gives an explicit geometric proof of completing the square in Al-Jabr, using a literal diagram of squares and rectangles
Muhammad ibn Musa al-Khwarizmi
1145
Robert of Chester translates Al-Jabr into Latin, introducing the geometric method of completing the square to European mathematicians
Robert of Chester
17th century
European algebraists translate the geometric procedure into symbolic form, enabling the general derivation of the quadratic formula

Summary

Completing the square rewrites ax² + bx + c as vertex form a(x-h)² + k.
The key step: add and subtract (b/2)² to turn x² + bx into a perfect square trinomial.
If a ≠ 1, factor a out of the x² and x terms before completing the square.
Vertex form directly reveals the parabola's vertex (h, k), which is a minimum if a > 0 and a maximum if a < 0.
Completing the square is the method used to derive the quadratic formula and to convert conic sections to standard form.

References

WebsiteWikipedia — Completing the square
Historical sourceAl-Khwarizmi, M. (c. 820 CE). Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala.
BookSullivan, M. Algebra & Trigonometry, Ch. 1: Quadratic Equations.