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Algebra · Algebraic proof

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Proofs with squares

The square-shaped proof toolkit: expanding (an + b)² honestly, reading parity out of 2(…) + 1, difference-of-two-squares shortcuts, and the completed square as an always-positive argument.

Algebra · Algebraic proof

Proofs with squares

The square-shaped proof toolkit: expanding (an + b)² honestly, reading parity out of 2(…) + 1, difference-of-two-squares shortcuts, and the completed square as an always-positive argument.

Why it works

Square proofs live or die on one expansion: (an+b)2=a2n2+2abn+b2(an + b)^2 = a^2n^2 + 2abn + b^2 — middle term included, every time. From there, three standard arguments do almost all the work.

Parity: exhibit 2(…)2(\ldots) or 2(…)+12(\ldots) + 1. Even means 2×2 \times whole; odd means one more than that. The square of any even number: (2n)2=4n2=2(2n2)(2n)^2 = 4n^2 = 2(2n^2) — even (in fact a multiple of 4). The proof's final move is always rewriting into the parity shape and saying which it is.

Difference of two squares: factorise instead of slogging. a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b) turns square-difference claims into short products. The difference between the squares of n+5n + 5 and n−5n - 5:

(n+5)2−(n−5)2=[(n+5)+(n−5)][(n+5)−(n−5)]=(2n)(10)=20n(n+5)^2 - (n-5)^2 = \big[(n+5)+(n-5)\big]\big[(n+5)-(n-5)\big] = (2n)(10) = 20n

— a multiple of 20, in one line. (Expanding both squares works too: same destination, more places to drop a sign.)

Always-positive: complete the square. A square is never negative, so (x−a)2+k≥k(x - a)^2 + k \ge k. To prove an expression is positive for ALL xx, complete the square and point at the floor. Substituting sample values proves nothing — the completed square covers every xx at once.

Counterexamples still rule disproofs. "n2+n+11n^2 + n + 11 is prime for every nn" survives n=1,2,3…n = 1, 2, 3\ldots — and dies at n=10n = 10: 100+10+11=121=112100 + 10 + 11 = 121 = 11^2. Formulas that LOOK prime-generating are a classic trap: hunt the counterexample where the formula's constant can appear as a factor.

Present square proofs like all proofs: general form first, honest algebra in the middle, and a closing sentence that names the property ("...which is 2×2 \times whole +1+ 1, so odd").