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Calculus IV
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Course: Algebra III / Intermediate Algebra
Topic: Nonlinear Equations and Applications
Subtopic: Quadratic Equations III - Quadratic Formula

Overview

This lesson covers the last of four methods for algebraically solving a quadratic equation. From the quadratic equation ax2+bx+c=0, identify the a,b,c, and plug into the quadratic formula , and simplify to give the exact x-solutions. The quadratic formula is a particularly useful method since it can be used to solve any quadratic equation, factorable or not. However, it is a bit tedious and lends itself to several common errors worth discussing.

The four different ways of solving a quadratic equation algebraically:

Study all of these methods carefully and know when to use which one!

Real solutions to a quadratic equation can also be found by graphing electronically (the x-intercept values are the Real solutions). Non-Real solutions can only be found algebraically.

A quadratic equation always has two answers. But, are they always different from one another? When are they Real vs. non-Real? The discriminant (radicand of the quadratic formula) can be used to answer these questions without having to fully solve or graph the quadratic equation.

Objectives

By the end of this topic you should know and be prepared to be tested on:

Terminology

Define: general quadratic equation, quadratic formula (memorize!), quadratic formula method of solving a quadratic equation, radicand, discriminant D=b2-4ac, roots of a (quadratic) equation

Text Notes

Your text shows the derivation of the quadratic formula. It is not necessary to understand this process, but it helps some people to know from where the formula comes -- it's not out of the blue! It is however important to know that one can complete the square on the general quadratic equation to solve it for x and the solution is the quadratic formula, thus the quadratic formula is really just a general form of the solutions to a quadratic equation.

Supplementary Resources

Read my Lecture - Applications of Quadratic Equations.

Read Plus Magazine's 100 Uses of Quadratic Functions - Part I
and 100 Uses of Quadratic Functions - Part II.