It is the everyday mental-maths shortcut for squaring numbers near a round figure. To square 23 quickly, write it as (20+3)²: 20² + 2·20·3 + 3² = 400 + 120 + 9 = 529 - no long multiplication needed. Engineers and builders use it when a square area grows: enlarge a square plot of side a by an extra strip b on two sides and the new area is exactly a² + 2ab + b², which is why a small increase in side length adds disproportionately more area (the 2ab cross-term). It also appears whenever you expand squared expressions in physics formulas, statistics (variance of a sum), and finance (compounding two combined rates).
Why is there a 2ab term - where does the 2 come from?
Geometrically, draw a square of side (a+b). It splits into four pieces: a small a×a square, a small b×b square, and TWO identical a×b rectangles - one along the top, one down the side. Those two rectangles are the 2ab. People who forget the middle term and write (a+b)² = a² + b² are forgetting those two rectangles, which is one of the most common algebra mistakes. The animation here shows exactly those terms building up.
How does this help with squaring numbers in my head?
Split the number into an easy part plus a small part. 31² = (30+1)² = 900 + 60 + 1 = 961. 52² = (50+2)² = 2500 + 200 + 4 = 2704. The trick works because the round part squares easily, the cross term 2ab is a quick doubling, and the small square b² is tiny. With practice this is far faster than long multiplication and is a favourite of mental-maths and competitive-exam techniques.
Does (a+b)² connect to anything in statistics or data?
Yes. When you add two quantities and look at how their combined value varies, the variance of a sum expands just like (a+b)²: Var(X+Y) = Var(X) + Var(Y) + 2·Cov(X,Y). The 2ab cross-term becomes twice the covariance - how the two move together. So the same algebraic shape that squares 23 also explains why combining two correlated investments or measurements does not simply add their individual variabilities.
Is this identity used in computing or graphics?
It shows up in distance and lighting calculations. Squared distances (used everywhere in graphics and physics to avoid slow square roots) expand using these identities. When a coordinate changes by a small amount, (a+b)² tells you how the squared distance shifts, and the cross-term 2ab is the dominant change for small b - the basis of many fast approximations in game engines and simulations.