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Quadratic Formula – Derivation

Standard 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

quad_deriv223.gif (1759 bytes)

Note that we multiplied everything under the radical by a2 and divided outside the radical by a. Rearranging this just a little gives us

quad_deriv224.gif (1134 bytes)

The Quadratic Formula

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Recommended Books

cover College Algebra (Schaum's Outlines)

    The classic algebra problem book - very light on theory, plenty of problems with full solutions, more problems with answers

 

cover Schaum's Easy Outline: College Algebra

    A simplified and updated version of the classic Schaum's Outline.   Not as complete as the previous book, but enough for most students

 

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