Why are square roots so fundamental?

Why are square roots (and squared quantities) so important in physics, mathematics, and all over science? Whether it's the area of a circle, the quadratic formula, the Pythagorean theorem or The Grand Design/The Theory of Everything (Hawking), this seemingly arbitrary X times X has implications everywhere! Why not (2x - 1)? Why not (x cubed minus x)?

I believe it rests in their perfect-square nature. The Greeks recognized their importance primarily because of how they illustrated quantities. In reference to square quantities, their numbering system illustrated them as simply squares; for instance 4 was represented by 4 dots arranged in a 2 by 2 square, 9 was a 3 by 3 square... This somewhat perfect nature in quantifiable number-sense that is, or perhaps this pattern, if you will, is significant in many aspects of our world... However, I wonder, if the ultimate flaw in our thoughts or prejudgment (at least until recently) may indeed be this obsession with finding order/patterns of repetition in our world?

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