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:
- Factoring into the form ( )( )=0 and using the zero-product rule, but only works if the quadratic is factorable
- Root method, but this only works when the quadratic is in the form (expression)2=#
- Completing the square, but this only works when the x2 term's coefficient is 1
- Quadratic formula is particularly versatile, memorize it!
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:
- 11.4.1 Recognize that all quadratic equations can be solved by the quadratic formula method
- 11.4.2 Solve quadratic equations algebraically by the quadratic formula method
- 11.4.3 Find the discriminant of a quadratic equation
- 11.4.4 Determine the type of solutions to a quadratic equation by graphing or using the discriminant
- 11.4.5 Understand the connections between the discriminant being zero/positive/negative, the effect on the graph of the quadratic function, the number of x-intercept points on the graph of the quadratic function, and the number of real/non-real solutions of the associated quadratic equation
- 11.4.6 Given two numbers, form a quadratic equation to which they would be solutions
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
Text:
Intro & Inter Algebra for CS 3ed by Blitzer, sect. 11.2
- pg 753 shows the derivation of the quadratic formula. It is not important to understand this process, but it helps some people to know from where the formula comes -- it's not out of the blue! This page shows how you 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.
- pg 755 Take note of the caution. Not doing so will cause you to make frequent errors while simplifying the answers from the quadratic formula.
- pg 757 discusses the connection between the graph of the quadratic function and the number of real solutions of the associated quadratic equation.
- pg 757-758 discusses the discriminant and the connections between the quadratic equations' discriminant, graph, number of x-intercepts, and the number of real/non-real solutions.
- pg 759 has a useful chart comparing the four methods of solving a quadratic equation and some tips for choosing the best method.
- bottom of pg 759 and example 5 works the "backwards" process of starting with given solutions and finding a quadratic equation that would have those solutions.
Supplementary Resources
Read 100 Uses of Quadratic Functions - Part I and 100 Uses of Quadratic Functions - Part II.