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 ax^{2}+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 x
^{2}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 (i.e., working "backwards" from the solutions to the equation)

**Terminology**

Define: general quadratic equation, quadratic formula (memorize!), quadratic formula method of solving a quadratic equation, radicand, discriminant D=b^{2}-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.