Posted Dec 15, 2019

Some have expressed concerns that the method I shared is exactly the same as the traditional method of Completing the Square. The algebraic manipulations of expressions are very similar. However, there is a difference in the logical steps, which is evidenced by differing requirements on initial assumptions. This is a logical distillation of Completing the Square as commonly learned:

- By adding and subtracting terms from both sides, and combining terms, the quadratic equation
is equivalent to a form for some specific expressions and in terms of and . This is equivalent in the sense that the two equations have the same set of roots. - The complete set of numbers that satisfy
is , and no other numbers square to . - Therefore,
if and only if is equal to one of . We then get the complete set of solutions to the original quadratic.

When compared to the method I shared, Step 2 above assumes that the complete set of square roots of every number is known, whereas Step 4 in the method I shared just needs one working square root.

Depending on the context, existence assumptions can be much easier to settle than complete-set assumptions. As an analogous example: it is easy to observe that

If one wishes to make Completing the Square more complete (at least to match the method I shared), then some of the simplest ways return to the notions of factoring and the zero-product property. For example, to show that the only numbers that square to