Completing the Square: Deriving the Quadratic Formula

If you haven't yet studied complex numbers (being numbers with an "i" in them), then you'd stop at this point, and say that this equation has "no solution". On the other hand, if you have studied complexes, you can continue.

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Again, if you don't yet know about complex numbers (the numbers with "i" in them), then you would say that the above quadratic has "no solution". If, on the other hand, you do know about complexes, then you would say that this quadratic has "no real solution", but it does have two "complex" solutions.

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Since solving "(quadratic) = 0" for x is the same as finding the x-intercepts (assuming the solutions are real numbers), it stands to reason that this quadratic should not intersect the x-axis (since x-intercepts are "real" numbers). As you can see below, the graph does not in fact cross the x-axis.

This relationship is always true. If you end up with a non-negative value on the right-hand side of the equation before you take the square roots of each side, then the quadratic will have two x-intercepts (or only one, if you get plus-minus of zero on the right-hand side); if you get a negative value on the right-hand side, then you won't get a real-number solution (because you can't take the square root of a negative within the real numbers), and the quadratic will not cross the x-axis.

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I'll do one last "example". It's a hard one, but please bear with me.

It has become somewhat fashionable to have students derive the Quadratic Formula themselves; this is done by completing the square for the generic quadratic equation ax² + bx + c = 0. While I can understand the impulse (showing students how the Formula was invented, and thereby providing a concrete example of the usefulness of abstract symbolic manipulation), the computations involved are often a bit beyond the average student at this point.

Here is what the instructor is looking for:

• Derive the Quadratic Formula by solving ax² + bx + c = 0.

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Whether you're working symbolically (as in the last example) or numerically (which is the norm), the key to solving by completing the square is "practice, practice, practice". By so doing, the process will become more "automatic", and you'll be much more likely to remember the steps when you're taking the next test.