Difference of squares a² − b²

Definition

The difference of squares of two real numbers a and b is the algebraic identity a² − b² = (a − b)(a + b). The left-hand side is the difference of two squares; the right-hand side factors the same polynomial as a product of two linear binomials. This calculator evaluates both paths: it multiplies (a − b) by (a + b), then shows the check a·a − b·b and a geometric reading — the area of a large square minus a small square (a frame).

The identity belongs to the compact multiplication formulas (alongside (a + b)² and (a − b)²). It holds in every commutative ring, in particular in ℝ and ℂ. MathWorld (Difference of Two Squares) and classical elementary-algebra texts give the same factorization. In school it shortens algebraic fractions, solves equations of the form x² − c² = 0, and supports mental arithmetic (for example 13² − 5² = 8 · 18).

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 “subtracting a square from a square” already appears in Euclid’s Elements (Book II — square and rectangle identities). Modern symbolic algebra wrote the same relation as a² − b² = (a − b)(a + b). In nineteenth–twentieth-century school algebra the formula entered the compact-multiplication canon; in analysis it appears when estimating (a − b)(a + b) = a² − b² in limit proofs, and in trigonometry (1 − sin² x = cos² x is a difference of squares in disguise).

Conceptually the calculator does not solve equations — it only evaluates the identity for given a, b. Do not confuse it with (a − b)² (square of a difference, three terms) or with (a + b)² (square of a sum). The difference of squares is a product, not the square of a binomial.

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: a² − b² − (a − b)(a + b) = 0. Antisymmetry under a ↔ b: the result changes sign. When a = b the result is zero. When b = 0 it reduces to a². Homogeneity: scaling a ↦ λa, b ↦ λb multiplies both sides by λ².

Link to the square of a sum: (a + b)² − (a − b)² = 4ab — an identity tying three compact formulas. In the polynomial ring ℝ[x], the polynomial x² − c² factors as (x − c)(x + c) by the same rule.

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. Write the product (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 = 0. Hence (a − b)(a + b) = a² − b².

Step 4 (geometry). For a ≥ b ≥ 0 a square of side a has area a². Inside it place a square of side b. The frame area is a² − b². Reshaping the frame into a rectangle with sides (a − b) and (a + b) recovers the right-hand side (the classical Elements II picture).

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: 5² − 3² = 16 = 2 · 8; 10² − 6² = 64 = 4 · 16; 13² − 5² = 144 = 8 · 18. When a = b: result 0. When b = 0: result a². When a = 0: result −b². Negative b: 5² − (−3)² = 25 − 9 = 16, and the product path is (5 − (−3))(5 + (−3)) = 8 · 2 = 16.

Swapping 3 and 5 gives −16: the sign is part of the result, not an error message. Decimals: 0.5² − 0.1² = 0.24.

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 notes

The calculator computes product = (a − b) * (a + b) and check = a·a − b·b in floating-point arithmetic and shows both in the panel. Extreme |a|, |b| may overflow or lose precision when subtracting close squares — the product (a − b)(a + b) is often more stable when a ≈ b. A comma and a period are equivalent.

The calculator does not factor a³ − b³ or a⁴ − b⁴ — only the difference of squares of two scalars.

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: square of a sum (a + b)², square of a difference (a − b)², one-number powers, the quadratic equation x² − c² = 0. In trigonometry, 1 − sin² θ = cos² θ is a difference of squares. In higher algebra — factorization in UFD rings.

Authorities: Euclid, Elements II; elementary algebra texts; MathWorld — Difference of Two Squares; Wikipedia — Difference of two squares. 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.