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Quadratic Formula – DerivationStandard Quadratic Form · The Quadratic Formula · Quadratic Theory and Graphs · Completing the Square · Recommended Books
Starting from Standard Quadratic Form
The most useful (but not always the easiest to use) formula for solving quadratics is the quadratic formula. In order to use it, the equation must be in the standard form
Quadratic Standard Form
Note that this standard form differs from that used with completing the square (CTS from now on), so it is important to work with the equations and their respective standard forms so much that confusion is unlikely. We will use completion of the square and a little substitution to derive the quadratic formula. First, divide the equation by a:
Next, substitute
And the equation becomes
This second version of the equation is in standard form for CTS (compare them and see how different they are). Plugging the appropriate symbols into the CTS formula gives
solving for x gives
Unsubstituting b and g gives
Note that we multiplied everything under the radical by a2 and divided outside the radical by a. Rearranging this just a little gives us
The classic algebra problem book - very light on theory, plenty of problems with full solutions, more problems with answers
A simplified and updated version of the classic Schaum's Outline. Not as complete as the previous book, but enough for most students