Completing the Square

Completing the square is one method for solving a quadratic equation. A quadratic equation can also be solved by several methods including factoring, graphing, using the square roots or the quadratic formula. Completing the square will always work when solving quadratic equations and is a good tool to have. Solving a quadratic equation by completing the square is used in a lot of word problems.

You guys are going to learn about lots of different ways to solve quadratic equations.

Graphing is great in finding the X intercepts except for sometimes it's not very precise, especially if you have an answer that's not a whole number. Also, quadratic equation is good but sometimes people make mistakes with the negative signs.

Completing the square will always work. Factoring doesn't always work. That's why completing the square is a good tool to have in your options for how to solve quadratic equations.

Before we start talking about how to do it, what I want to do is review what you guys already know about perfect square trinomials. They look like this. Perfect square trinomials look like A plus B equals A squared 2 A plus B squared or A minus B squared equals A squared minus 2AB plus B squared. This is our goal.

Completing the square means taking some trinomial and writing it like this as a squared binomial. Like, for example, if we have X squared plus BX, we can complete the square or turn it into what would be a factored perfect square binomial by adding B over 2. Squared. That's tricky.

It's going to make a lot more sense once you guys start trying these actual practice problems. But if I can take half of this B term and then square it and add it to this original statement, I would have a perfect square trinomial. And we'll do some package problems to have it make more sense.

Here's the step-by-step process for how to complete the square. First thing you want to do is rewrite the trinomial in the form X squared plus BX equals C. It's super important that you notice that my leading coefficient here is 1. If my number there was like 4 or 10 or negative 6 or something I'd have to divide all three terms by that number. It's absolutely critical that your leading coefficient is 1. So you get your X squared term and your X to the first power alone on one side of the equals and your constant goes on the other side.

The next thing you're going to do is take half of your B term, square it, and add the results to both sides of the equation. Like what I would do here is take half of B. B over 2. Square that quantity and then add it to both sides of this equation. After it's been squared.

So what I'm going to have is X plus half of B squared is equal to C plus half of B squared. That will make a lot more sense once we start working with real numbers. I just wanted to show you what this looks like with symbols.

The next thing I would do to finish completing the square or to solve for X would be to take the square root of both sides. So I would square root here and square root here and then go through and solve for X.

It looks really tricky here when we're just looking at the formulas and symbols, but once you guys start trying some problems that have real numbers it will start making a lot more sense for you.