The algebraic identity for a minus b whole square is:

(a−b)2=a2−2ab+b2(a-b)^2=a^2-2ab+b^2(a−b)2=a2−2ab+b2

Quick Scoop

1. The core formula

  • When you see “a minus b whole square”, it means (a−b)×(a−b)(a-b)\times (a-b)(a−b)×(a−b).
  • Expanded, it always becomes:

(a−b)2=a2−2ab+b2(a-b)^2=a^2-2ab+b^2(a−b)2=a2−2ab+b2

  • Pattern to remember:
    • First term: square of the first →a2\rightarrow a^2→a2
    • Middle term: minus twice the product →−2ab\rightarrow -2ab→−2ab
    • Last term: square of the second →b2\rightarrow b^2→b2

2. How it is derived (FOIL idea, in words)

Imagine multiplying the bracket by itself:

  • First: a×a=a2a\times a=a^2a×a=a2
  • Outer: a×(−b)=−aba\times (-b)=-aba×(−b)=−ab
  • Inner: (−b)×a=−ab(-b)\times a=-ab(−b)×a=−ab
  • Last: (−b)×(−b)=b2(-b)\times (-b)=b^2(−b)×(−b)=b2

Combine the middle terms: −ab−ab=−2ab-ab-ab=-2ab−ab−ab=−2ab, giving a2−2ab+b2a^2-2ab+b^2a2−2ab+b2.

3. A quick numeric example

Take a=5a=5a=5, b=3b=3b=3:

  • Left side: (5−3)2=22=4(5-3)^2=2^2=4(5−3)2=22=4
  • Right side: 52−2⋅5⋅3+32=25−30+9=45^2-2\cdot 5\cdot 3+3^2=25-30+9=452−2⋅5⋅3+32=25−30+9=4

Both sides match, so the identity holds.

4. Why this identity is useful

  • Simplifying algebraic expressions: e.g., (2x−5y)2=4x2−20xy+25y2(2x-5y)^2=4x^2-20xy+25y^2(2x−5y)2=4x2−20xy+25y2.
  • Mental math: compute things like 99299^2992 by treating it as (100−1)2(100-1)^2(100−1)2.
  • Appears in many topics later: quadratic equations, coordinate geometry, and more.

5. Tiny “story” to remember it

Think of (a−b)2(a-b)^2(a−b)2 as:

“Start with the big square a2a^2a2, remove two rectangles of area ababab, and keep the small square b2b^2b2.”

That mental picture matches the terms a2−2ab+b2a^2-2ab+b^2a2−2ab+b2. TL;DR:
“a minus b whole square” means (a−b)2(a-b)^2(a−b)2, and the expansion is always

(a−b)2=a2−2ab+b2.(a-b)^2=a^2-2ab+b^2.(a−b)2=a2−2ab+b2.