// maths › Square Identities
Expand and verify (a−b+c)² with its specific cross-term signs, shown numerically.
(a-b+c)² = a²+b²+c²-2ab-2bc+2ca
A mind behind this: Al-Khwarizmi c. 780–850 CE
In any net calculation of the shape 'start with a, subtract b, add c, then square the result' - then squared for an energy, error, or distance measure. Here a and c are positive and b is negative, so the a-c cross-term is positive while the a-b and b-c terms are negative. It is the mirror of (a+b−c)² with the roles of b and c swapped, and shows up wherever the middle quantity is the one being subtracted.
Use the sign-product rule: a pair's cross-term sign is the product of the two variables' signs in the bracket. For (a − b + c): a·b = (+)(−) = − → −2ab; b·c = (−)(+) = − → −2bc; c·a = (+)(+) = + → +2ca. So the two pairs that include the subtracted b are negative, and the a-c pair is positive. The squared terms are always positive. This single rule replaces memorising every variant.
Yes, in resultant and residual calculations where the middle of three contributions opposes the others - a force or displacement that points the opposite way, then the magnitude is squared. It also appears in coordinate geometry when a point's signed offsets combine and you compute a squared distance. The specific sign pattern matters for getting the magnitude right.
Very much. Questions often present a six-term expression and ask you to factorise or simplify it. Spotting that a²+b²+c²−2ab−2bc+2ca is exactly (a−b+c)² turns a daunting expression into one tidy bracket, saving time and reducing mistakes. Reverse-recognition - expansion to factor - is a frequently tested and genuinely useful skill.
Because seeing each signed variant verified numerically cements the underlying sign rule far better than reading them in a list. Plug in your own numbers, watch each term's sign and value appear, and confirm both sides match. After working through the variants, you stop memorising and start deriving them - which is the real goal.