Where would I square a linear expression like ax+by+c?
All the time in optimisation, statistics, and physics, because squared linear expressions are how we measure error and energy. A line or plane ax+by+c describes a relationship; squaring its value gives the squared deviation from a target, which is exactly what least-squares fitting minimises - the method behind trend lines, regression, and most of machine learning. The expansion shows every contribution to that squared error, including the cross-terms that couple the variables.
Why are there six terms - three squares and three cross-terms?
Because squaring a three-term expression pairs every term with every other. The three terms ax, by, c each get squared (giving a²x², b²y², c²), and each of the three pairs produces a doubled cross-product (2abxy, 2bcy, 2cax). It is the same structure as (a+b+c)² but with the coefficients and variables carried along. Six terms is simply 'three squares plus three pairs'.
How does this relate to least-squares and regression?
Fitting a line or plane to data means choosing coefficients to minimise the sum of squared residuals, where each residual is a linear expression like (ax+by+c) − observed. Expanding those squares produces exactly these square and cross terms, and the cross-terms are what make the variables interact in the solution. The normal equations of regression come directly from expanding and differentiating such squared linear forms.
Does it appear in physics or engineering?
Yes - the squared magnitude of a quantity built from several linear components expands this way. Energy often depends on the square of a linear combination of variables (displacements, currents, fields), so computing total energy involves exactly these squared and cross terms. The cross-terms represent interaction or coupling between components, which is physically meaningful - they show how the parts reinforce or oppose each other.
Is recognising this pattern useful for simplification?
Very. A sprawling expression with three squared terms and three matching cross-terms can be collapsed into a single squared bracket (ax+by+c)², which is far easier to handle. Spotting this in an engineering derivation or exam question turns six terms into one, reducing the chance of error and often revealing structure (like a perfect square) that suggests the next step.