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The Balylonians (around 400BC) first developed the algorithmic method for completing the square. But they didn't do it using equations, which they had no concept of, just a set of rules for particular cases. The Greeks, such as Euclid, showed geometrical proofs of the method. But it wasn't until the Persian mathematician al-Khawarizmi in the 9th century that the general algorithm was written down. Even this wasn't done using symbols like we use in algebra today but written out in prose.

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Q: Who invented the principle of completing the square?
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