So you've just learned all possible methods of solving a quadratic equation, but that means you can be given a quadratic equation with no direction on what method to use and just be expected to choose the best one. But how do we know which method is best for any given quadratic? Well, we've been collecting information on when we should use each method, and here, I'm going to use that information to show you exactly how to choose what method is best. It's important to note here that multiple methods may work for any given quadratic, but here, we're focused on choosing which one is best. Which one is going to make our quadratic equation the easiest and most straightforward to solve? So let's go ahead and get started. In our table here, we're going to look at our use if column and a structured way to go through this is to start with factoring and go through each method until you find one that matches up with your quadratic equation. As you get more practice, you're going to be able to just look at a quadratic and you might be able to see right away which method it matches up with, and that's fine as well. But let's go ahead and start with some structure.
Looking at our first example here, we want to choose which method is best, and I have \( x^2 + 3x = 0 \). First, let's check my factoring box to see if any of this criteria matches. It either has obvious factors or \( c \) is equal to 0. Here, I do have \( c \) equal to 0 because I have no constant. So I can actually stop here because I know that factoring is going to be the best method to use. Let's go ahead and take a look at our second example. Here I have \( x^2 + 6x + 1 = 0 \). Let's again start with our factoring box. Does this have obvious factors, or is \( c \) equal to 0? Well, it doesn't have obvious factors to me, and I do have a constant, I have this 1. So factoring is not going to be the best choice here.
Let's move on to the square root property. Does it have this form \( x + \text{some number}^2 = \text{a constant} \)? No. And is \( b \) equal to 0? Also no here because I have a \( b \) of 6. So I'm not going to be able to use the square root property either. Let's move on to completing the square. Is my leading coefficient 1 and is \( b \) even? Well, I do have a leading coefficient of 1 because that \( x^2 \) is just by itself there, and \( b \) is 6, which is an even number. So that tells me that completing the square is going to be the best choice to solve this one.
Let's look at another example here. We've been able to kind of go through each of them and see which one works. So here I have \( (x+2)^2 = 9 \). This is an example of a quadratic that we're going to be able to look at and know exactly which I know is one of the clues that I should use the square root property. So here, the square root property is going to be the best method.
Now we just have one more left. Let's take a look. So I have \( 2x^2 + 7x + 3 = 0 \). Let's go ahead and go through left to right one more time. So does this have obvious factors? Well, definitely not. And \( c \) is not equal to 0, it's equal to 3, so I'm not going to want to factor here. Moving on to the square root property, it definitely doesn't have this form \( x + \text{a number}^2 = \text{a constant} \), and I do have a \( b \) because I have this 7 \( x \). So not the square root property either. And completing the square, is this going to be the best method? Well, my leading coefficient is not 1, so completing the square is already out of the picture, which just leaves me with the quadratic formula. So that is going to be the best method here. Remember, again, multiple methods can work for any quadratic, but these are the best ones. And that's how to choose the best method for any given quadratic equation. Let me know if you have any questions.
