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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 **c**ompleting
**t**he **s**quare (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 a^{2} and
divided outside the radical by a. Rearranging this just a little gives us

College Algebra (Schaum's Outlines)

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

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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