Finding Complex Roots of Quadratic Equations

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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will see how to find complex roots of quadratic equations using the quadratic formula. We will also look at what the graph of a quadratic equation with complex roots looks like.

Complex Roots

Let's suppose you are reading a tabloid, and you see a blurb in the table of contents that says that the popularity of a starlet in the media (based on the number of articles being written about the starlet) seemed to be going down for a while, but then there was some sort of scandal, causing the starlet's popularity to slowly rise again.

This gets you to wondering if there was ever a time when the starlet's popularity was non-existent during this time. In other words, was it ever at a point where no articles were being written about this person? You see that the blurb says that the starlet's popularity, over the course of the last month, could be modeled by the equation P = x 2 - 30x + 229, where P is the number of articles that mention the starlet, and x is day number over the past 30 days.

You're a little confused as to why they put this random math formula in the article, but you're glad they did, because you can use it to answer your question! To find out if, at any point, the starlet had no mention in any articles, you simply plug in 0 for P and solve.

x 2 - 30x + 229 = 0

First off, we notice that the highest exponent in this equation is 2. This tells us that this is a quadratic equation. Quadratic equations are equations in which the highest exponent of the variable is 2. Secondly, we notice that in solving this, we are finding the values of x that make P zero. There is a name for these values, and that is the roots of the equation, since they are the values that make the equation true.

You're not sure how to solve this, so you just take the easy way out and use your graphing calculator's solve function. The calculator says that the solutions, or roots, of this equation are x = 15 + 2i and x = 15 - 2i. Wait, these are complex numbers! Complex numbers are numbers of the form a + bi, where i = âˆš-1.

So, what does this tell us? Well, since the only roots are complex numbers, there are no real numbers that satisfy the equation. This tells us that it can't be the case that the starlet ever got to the point of not being mentioned in any articles over this 30-day period.

Well, that's pretty neat! We can find out a lot about a situation that's modeled by a quadratic equation when that equation has complex roots. You decide you want to explore finding complex roots of quadratic equations further. I don't blame you!

What if you hadn't had your calculator handy to solve this problem? Well, you figure you could have graphed P = x2 - 30x + 229 by hand and analyzed the graph.

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