Square of a sum (a + b)²

Definition

The square of a sum of two real numbers a and b is the algebraic identity (a + b)² = a² + 2ab + b². The left-hand side is the square of a binomial; the right-hand side expands the product (a + b)(a + b) by distributivity. This calculator evaluates both sides and shows the terms a², 2ab, b² together with a geometric reading: the area of a square of side a + b split into a×a, b×b, and two a×b rectangles.

The identity belongs to the compact multiplication formulas (alongside (a − b)² and a² − b²). It holds in every commutative ring, in particular in ℝ and ℂ. MathWorld (Perfect Square) and classical elementary-algebra texts give the same expansion.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Historical and conceptual context

Geometric “completing the square” was known in Babylonian mathematics and in Euclid (Elements II). Symbolic algebra of the sixteenth–seventeenth centuries (Viète, Descartes) wrote it as (a + b)². The formula is a step toward solving quadratics by completing the square.

Conceptually the calculator does not solve equations — it only evaluates the identity for given a, b. Do not confuse it with (a + b²) or a + b²; the parentheses enclose the whole sum before squaring.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Formal properties

Domain: a, b ∈ ℝ (or ℂ). Identity: (a + b)² − a² − 2ab − b² = 0. Variants: (a − b)² = a² − 2ab + b²; a² − b² = (a − b)(a + b).

Homogeneity: scaling a ↦ λa, b ↦ λb multiplies both sides by λ². Symmetry: swapping a ↔ b leaves the result unchanged. For b = 0 it reduces to a².

In an inner-product space ||u + v||² = ||u||² + 2⟨u,v⟩ + ||v||² — the same pattern in Euclidean geometry.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Full derivation

Step 1. (a + b)² := (a + b)(a + b).

Step 2. Distributivity: a·a + a·b + b·a + b·b = a² + ab + ba + b².

Step 3. Commutativity of multiplication: ab = ba, so ab + ba = 2ab. Hence (a + b)² = a² + 2ab + b².

Step 4 (geometry). A square of side a + b (for a, b > 0) splits into an a×a square, a b×b square, and two a×b rectangles — summing areas recovers the right-hand side (Euclid II.4).

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Special cases

Examples: (3 + 4)² = 49 = 9 + 24 + 16; (5 + (−2))² = 9 = 25 − 20 + 4; (1.5 + 0.5)² = 4. When a = b: (2a)² = 4a² = a² + 2a² + a².

For a = −b: (a + b)² = 0, and the right-hand side a² − 2a² + a² = 0. Negative side lengths in the diagram are read algebraically (signed areas), not as physical lengths.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Numerical and computational notes

The calculator computes left = (a + b) * (a + b) and right = a·a + 2·a·b + b·b in floating-point arithmetic and compares them. Extreme |a|, |b| may overflow. Comma and period are equivalent.

The calculator does not expand (a + b + c)² — only two summands.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Related concepts and authorities

Related: maximum of two numbers, linear interpolation (different geometry), the exponential (non-algebraic). The companion school identities are (a − b)² and the difference of squares.

Authorities: Euclid, Elements II; elementary algebra texts; MathWorld — Perfect Square Binomial; Wikipedia — Polynomial identities. Educational tool.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.