Posted Dec 15, 2019

Quadratic Method: Completing the Square

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:

  1. By adding and subtracting terms from both sides, and combining terms, the quadratic equationis equivalent to a formfor some specific expressionsandin terms ofand. This is equivalent in the sense that the two equations have the same set of roots.
  2. The complete set of numbers that satisfyis, and no other numbers square to.
  3. Therefore,if and only ifis 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 thathas at least one solution, because. Many students initially think that there is only that one cube root. Only later do they find out thatalso have cubes equal to 8. So, the assumption that there are no other complex numbers that square toother thanactually requires further justification.

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 toare, it suffices to find all solutions to. This is equivalent to:This can be factored by finding numbers that sum to 0 and multiply to. Selectingworks. Therefore, this equation is equivalent to:and by the zero-product property,are all the solutions.


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