 Hello everyone and welcome to Tutor Terrific. Today we are going to look at part 3 of our Solving Quadratic Equations series. It's the last part and these are the methods we are going to use when factoring is not possible or does not appear to be possible by the means in the first two parts of the series. The three types of methods we are going to look at today are solving by square roots, completing the square and the quadratic formula. All of these, when factoring is not possible, will result in solutions that are irrational. So that means they have a square root in them. So the decimal form is an irrational number where the decimal does not repeat and it goes on forever. Alright, let's get started. First method is solving by square roots. Now you can use the square root method in all different types of situations, including instead of the difference of squares method that I showed you in part 1 of this series. However, square roots is necessary when you cannot solve a difference of squares type binomial with the difference of squares factorization. Sometimes you just have to square root. So the premise here is that you use this method when the x to the first power term is missing and you will move the constant terms to the other side and make sure all the x squared terms are on the same side. Then you'll simplify the expression if necessary and then square root. You will get two answers by doing this and I'll show you how. Let's look at the first one. Notice how it's often that solving by square roots problems are not in standard form and that's fine. You don't want them to be in standard form. You want the x squared terms on one side and the constant terms on the other. So when we look at this first problem, I'm gonna add one to both sides to get started. Now I have x squared equals 18. Okay, now there's no simplification that I have to do. So what I'm gonna do now, I'll write it over here so it's easier to see, I'm going to square root both sides. Okay, now when I square root 18, I'm going to make sure whenever I square root a constant term in this method, I'm going to have to put a plus or minus. Why do you have to include the plus or minus? Well, let's consider, for example, the square root of nine. Okay, the square root of nine is three. But let's think about this really carefully. Couldn't it also be negative three? Well, negative three times negative three is nine and positive three times positive three is nine. So when you think about it, both negative threes and positive threes multiply by each other to get positive nine. And so that's why we include the plus or minus so that we can include all possible ways to get 18. Okay, now obviously the square root of x squared is x. But let's look at 18. Now 18 doesn't have a perfect square root, but can part of this be pulled out of the square root factored out. 18 is nine times two. I could stop here because nine is a perfect square of three. And so I could pull out the nine square root as I pull it out and make it a three. So I need to get three square root two plus or minus a positive root three, sorry, positive three times root two, and negative three times root two are the solutions to this quadratic. Now let's try this more complicated one. As you can see, I've got x squared terms on both sides, and I have constant terms on both sides. I'm going to simplify that first by looking at all the terms that all have a negative sign. If I multiply through by a negative one, that means multiply everything by negative one, I will get all those negative signs to go away. And you are totally allowed to do that. So now I multiply through by negative one, and I have everything positive. First I'm going to move this constant term to the other side. I'm going to move the x squared term to the other side as well. Now I have two x squared equals six. Okay, that's a lot simpler. Then I will simplify. In this case, I have to simplify, and I can see I can divide by two here, and I get x squared equals three. Okay, this is ready to square root. So up here, just for good measure, I'm going to create some space here. I'm going to square root both sides, and next to the constant term, I'm going to put a plus or minus because the reason I made clear above. Now let's look here. x will now equal plus or minus root three. Okay, so that required a little bit of extra work, but this is the basic premise for solving by square roots. Absolutely necessary when you have an x squared term and a constant term if you cannot factor by the difference of squares method. All right, now here's our second method completing the square. Now the premise of this method is that you have a quote unquote bad constant term which doesn't allow you to factor meaning I cannot find two numbers that multiply to the constant term and add to the coefficient of the middle term. So we will do is we'll move the bad constant term to the other side, and then we will add the proper constant term so that we can make a perfect square trinomial. What's a perfect square trinomial? Well, it's a trinomial that factors into the same binomial twice. Okay, it's like x plus three times x plus three. Okay, that makes x squared plus six x plus nine. So that's a perfect square trinomial example. We're going to make those by completing the square. So let's apply that to this first example. So what we have here, x squared plus four x minus seven, you could try and factor that good luck. I'm not going to find two numbers that multiply to negative seven and add to positive four because seven is prime. It's not possible. We don't even have to think about it. Okay, so what we're going to do is we're going to add that seven to the other side. So now we have x squared plus four x now leave a space, a big space and put equals seven in that space. What you're going to put is plus blank minus blank. The reason I have to do that is because I can't add something to one side unless I subtract it also. Alternatively, which I like to do as well, is put a plus blank on this side and a plus blank on the other side. Both of these are equivalent. Okay, now, the reason I want to do it this other way is because of the next example. So I'm going to do it where I put the plus and the minus on the same side. What goes in those blanks? Well, it goes in those blanks is half of the middle term squared. That is how you create a perfect square trinomial. You divide the middle coefficient four by two, which gives you two. And then you square it. And that gives you four again. In this case, it just happened to be the same as the four. So half the middle term squared is what you put in the blanks plus four minus four. Now you say, well, I just cancel those and they're gone. Now I'm back to where I started. No, what you're going to do is you're going to take the negative version and you're going to add it to the other side. Okay. So when we do that, we have the following and I'll write it up here. X squared plus four X plus four equals seven plus four, which is 11. Okay. Now, on the left, we have a perfect square trinomial. And what we're going to do with that is we're going to factor it. Now you don't have to think about how this would factor. It's going to factor the same way every time. It is going to factor like this. X plus two times X plus two. Now that's just the two comes from half of the middle term. You will add the half of the middle term to X and that will be both of your factors. When I have the same factor twice, I could just put one of those factors squared. And that equals 11. Now we already learned in this video this method of square roots. We are going to apply it now because I cannot move forward until I take the square root of both sides. Now remember, we're going to square root a constant. We have to put it as plus or minus the square root of the constant. On this side, we have X plus two squared square rooted. That just gives us X plus two. On the other side, we have this radical plus or minus square root of 11. I can't even take anything out of that. It is completely set like that. And the last step is just move the two to the other side by subtraction. We're not allowed to combine it with the square root any other way than this. X equals negative two plus or minus the square root of 11. So that's our answer to the first one. See how it has a radical in it just like all our answers from the previous method. Next, we're going to look at this guy. Now this one, the first coefficient is negative two and it doesn't look like it can divide that negative two by GCF grades common factor out of all the terms. So what I have to do is I have to factor it out of the first two, move the 31 to the other side first. So now I have negative two X squared plus 16 X equals 31. Alright, now I'm going to factor the negative two out of these two terms. You have to do this. You cannot skip this step. Negative two divided out gives us X squared minus eight X. Now I'm going to leave a space plus blank minus blank. Close the parenthesis equals 31. Notice how that's different than the previous step. There's added work here. Now we will complete this square inside the parentheses. We will divide negative eight by two. We get negative four and we square that. We get 16. So we will add 16 and subtract 16 next to each other. Now the negative 16 like the negative four up here we want to move to the other side. We can't just move it over there by adding it. We have to multiply it by what I call the gatekeeper. What was factored out of everything on this side in the beginning. So it's like negative 16 wants to go over here and the gatekeeper is like, hey, you got to multiply by me first. So when negative 16 multiplies by negative two we get positive 32. Okay, so we've got negative two X squared minus eight X plus 16. Now when I pulled this out it's plus 32 equals 31. What I'm going to do now is I'm going to subtract it to the other side. Negative two X squared minus eight X plus 16. Now we're going to subtract this 32. So it's over here on the other side negative 31 minus 32. What's that? Negative one. Negative one. Okay, now we need to simplify. Okay, let's get that negative two out of here and that's no problem. What we're going to do is divide both sides by negative two. That gets rid of the negative which would have been a huge problem when it was time to square root. So now we have one half over here. What do we have on this side? Well what we have is a perfect square trinomial. What we're going to end up doing with that is writing it as two binomials that are the same type so we can square one of them. Again it's half of the middle term added to X, the middle coefficient added to X that would be negative four. Aha, there we go. Now we will square root both sides. Plus or minus must be included over here. Then I have, say square root something squared it's just the thing inside, X minus four equals plus or minus square root one half. Then we will add the four to the other side. So our final answers will be that X equals four. Notice how I'm always putting the rational number first then plus or minus the irrational number. One half. Now I know there's multiple ways to write the square root of one half. It's not necessary in this video to discuss the various types. So there we go. X equals four plus or minus the square root of one half. This is the method of completing the square. Alright guys and here's our last method, the infamous quadratic formula. This can be used no matter what if you could factor it, if you can't factor it, but it's annoying and it can be quite a pain to simplify the expression you have to simplify, which is over here. And so some people avoid this and use it at last resort. However, the completing the square method does the exact same thing. Some people prefer the quadratic formula because they can remember that expression rather than the completing the square method. It can memorize it easier. So I'm going to do this first example, which is the same as my first example from the previous method, completing the square. Now we don't have to move anything around. We just want to get things in standard form and this expression is in standard form. What we have to determine is what a b and c are a of course is the coefficient of the first term. So our a in this example is one b is the coefficient of the middle term x and that's four. And then C is the coefficient of the constant term negative seven. So here we have our constants. Now we just plug them into the formula. It's very straightforward. That's why some people like it. So negative b that'll be negative four plus or minus the square root of b squared, which is 16 minus four a c. So that's four times one times negative seven. So that's minus negative 28 or plus 28. Okay, I multiplied those together. That's what I got. Then divided by two a all divided by two a. So that's two times one. That's two. Okay, now our only job is to simplify this expression as much as we can. So let's start with the what's called the discriminant. This b squared minus four a c is called the discriminant. We'll simplify that to one number. What's 16 plus 28 people looks to me like 44 negative four plus or minus the square root of 44 over two. Then we will try and pull out everything we can from the square root. And in this case, 44 is four times 11. I can pull out that four. I have to square root it while I do that. And so I get two. And then what's still inside is 11. So I have negative four plus or minus two root 11 over two. Now the next thing most people do is split this into two fractions and simplify further. So I got negative four over two plus or minus two root 11 over two. And you can see that those twos are going to cancel. And this first term is going to simplify to two over one, which gives us our result from the last step, the last method when we did this problem negative two plus or minus the square root of 11. So that's as you can see gives you the same answer as completing the square. Next, we're going to try this much more scary looking one. But it's really not that much scarier because we will just determine what A, B, and C are and plug them into the formula. So first things first, it's not in standard form because of this constant. So I'm going to add that to the other side because it's negative 10. I'm going to add 10 to both sides. So now I get 2x squared minus 7x minus three. Try as you might, you're not going to be able to factor that. So we can either complete the square or you can use the quadratic formula. That's what we're going to do here. A, the coefficient of the first term is two. B, the coefficient of the middle term, the x term is negative seven. And C, the constant term is negative three. So let's plug those in. So x is going to equal negative B. That's positive seven plus or minus the square root of negative seven squared. That's 49. You're squaring the negative. That's why you always get a positive there. Minus, then we have four times A times C. So four times two times negative three. What's four times two? Eight. What's eight times negative three? Negative 24. So again, we get a positive 24 over all over two A. That would be two times two. That would be four. Okay. So this looks like it's going to be a nasty fraction. That's fine. So we have seven plus or minus the square root of 73 over four. You can leave your answer like this if it's more simple. And I believe it is because if you were to split this up into two fractions, we couldn't simplify anything. We'd have seven fourths plus or minus the square root of 73 over four. Either of these two answers are correct. So there's no need to bicker about which one's better. They both are equivalent. So there is the quadratic formula. Straightforward, but rather annoying to simplify. So it's your choice when you use this or completing the square. All right, guys. Thank you so much for watching my series on solving quadratic equations. Tune in for more videos soon. This is Falconator signing out.