Proving (a+b)² ≥ 4ab

Intro to Proofs · Axiom Academy

How choosing the right starting point turns an algebraic proof into a few clean steps. Prove that for all real numbers a and b . The trick: instead of attacking the goal head-on, we start from a fact that is obviously true and steer it toward the result. Nice work — you proved an important algebraic inequality. Here's what made the strategy click: Strategic starting point: Beginning with was elegant because it is universally true and its expansion already contains the a^2 , b^2 , and ab terms we needed. Algebraic manipulation: Expanding, adding 4ab , combining like terms, and factoring transformed one inequality into the one we wanted. Perfect-square recognition: Spotting that a^2 + 2ab + b^2 = (a+b)^2 was the move that finished the proof. Equality condition: Equality holds exactly when a = b , since that is the only time (a-b)^2 = 0 . AM–GM connection: This inequality is a form of the Arithmetic Mean–Geometric Mean inequality, a foundational result across mathematics. We could have expanded (a+b)^2 directly and tried to show it is , but then we'd still have to justify . Starting from builds that justification in from the first line — choosing the right foundation makes a proof feel inevitable.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.