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Overview of the Different Methods of Solving a Quadratic Equation - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Solving quadratic equations can be difficult, but luckily there are several different methods that we can use depending on what type of quadratic that we are trying to solve. The four methods of solving a quadratic equation are factoring, using the square roots, completing the square and the quadratic formula.

So what I want to talk about now is an overview of all the different ways of solving a quadratic equation. What I mean by that is anything of the form: ax² plus bx plus c. So we have four different ways at our convenience. We have factoring, square root property, completing the square, and the quadratic formula. We can use these methods at different times, and what I want to do is just talk about when we can use them, why they're good, and why they're bad. So I'm just going to go down the row and talk about each one. The 'check' means pros and the 'minus' means cons. Factoring is typically the fastest and easiest way of solving something when it's factorable. Oftentimes, we're dealing with a quadratic that is not factorable, so then factoring is not going to help us. So it's fast and easy when it's usable, but not always factorable, either. So fast and easy, but not always applicable.
The next one we're going to talk about is the square root property. This is when we have something squared. So, the pro: is it's great when you're solving for something squared. The only problem is that it's not always the situation we're dealing with. Any time you have an X term or something like that we're not going to be able to use it. So it's not always a square term. It's great when applicable, but it's not always the case. It actually isn't the case very often at all.
Completing the square. The great thing about completing the square is we can always do it. There will never be a time you won't be able to complete the square. But the downfall is that it can get ugly. If you're dealing with a coefficient or an odd middle term or something like that you're going to introduce fractions. It's not always going to be the nicest situation.
Lastly, is the quadratic formula. It's great, again, because you can always use it. And cons, it depends on the person. If you're using square roots, which some people don't always like, you always have to use square roots as well. It's typically not as easy as some of these other methods, completing the square, I would say, is a little bit easier than that but it's something you have to remember. So you have to remember the formula, and it can get ugly.
So those are the four different ways, the pros and cons, and some things to think about when you're solving a problem. I'm not actually going to solve any for you. I just made a little chart so you'll know the resources you have available, and the pros and cons of each one.

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