 Hello everyone and welcome to Tutor Terrific. Today, in this video, I'm going to start a three-part series on solving quadratic equations. And this first part is the basic factoring methods you can use to solve quadratic equations. I'm going to go over three types in this video. Greatest common factor type factoring, which is what GCF stands for. Greatest common factor. The simple diamond method used for trinomials and the difference of squares used for quadratic equations that are the form of two squares with a minus sign in between them. So let's get started. Now the first type of factoring we're going to look at is called the greatest common factor type factoring. Now the premise of this type of factoring is that you're going to divide out and separate all factors common to all terms. So that's what's meant by greatest. We get all of them out. The largest amount of factors that belong to both or all the terms involved, we are going to divide those out and put them in front. Now of course this video assumes you have practice doing greatest common factoring with two or more terms. So let's look at this quadratic expression real quick. Expression versus equation because it's not equal to anything, it's just listed by itself without an equal sign. Now let's look here. We've got 4x squared plus 8x plus 16. Do you notice any factors that are common to all three terms? Well let's look at the x's. Do each of the terms have a power of x? No. The last term does not. So that x or x squared that you see in two of the terms is not a common factor because it does not occur in all the terms. But you do notice the 4, the 8 and the 16. So what's the largest number that is a factor of all three of these? If you break 4 up, it's 2 times 2. If you break 8 up, 4 times 2 and that 4 can break into 2 times 2. The 16 will break into 4 times 4 and each of those 4s will break into 2 times 2. Now it's the only time I'm going to do this because I remember I already assumed you've practiced doing greatest common factoring with numbers. You can see that each of these numbers has at least two 2s as its factors. So what we can do is we can pull out by division those 2s. Multiply together, you can pull out a 4. And so we can take out through division a4 and we're separating it in front. That's how this is done. When you divide the first term by 4, you get just the x squared left. When you divide 8x by 4, you get 2x. And when you divide 16 by 4, you get 4. And so here we have this expression factored. Now of course if it's an equation, it's usually set equal to 0. This is the most common way we solve these and we will use another method to continue to factor this. But we started with greatest common factor. Now let's look at the second one, which is not a trinomial with three terms. It's a binomial with two terms. As you can see, we have 15x squared plus 75x. So let's look at all the common factors. Let's look at the variables first. x squared and x. Do they share a common factor involving x's? Yes, they each have at least one x as a factor. So we can pull that out. So we know part of the greatest common factor is an x. Look at the numbers, 15 and 75. Is 75 a multiple of 15, just like 8 and 16 were multiples of 4? If so, then you can pull out 15. You can divide out 15. That would be the greatest common factor. And in fact, 75 is a multiple of 15. It's 15 times 5. So we divide out by division 15x. Now we write in parentheses what's left in our two terms of our binomial. If we divide 15x squared by 15x, we just get x. If we divide 75x by 15x, what would we get? We get 5. Okay? So this is what we'd have. And if it was set equal to 0, we would of course be able to solve this. Now remember the zero product property, which can be used right now in this example. Zero product property says that each factor, if the whole expression is set equal to 0, each factor can individually make it 0 and we can set each factor equal to 0. So we'll do that right now. 15x equals 0. That's the first factor. The second one is x plus 5. And you no longer need the parentheses when you write factors individually. If you solve both of these for x, you get the first one that x is 0. And the second solution for solving this quadratic equation, if you subtract 5 from both sides, you will see that x equals negative 5 also. So we have two solutions. We will always have two solutions for quadratic equations. That's how greatest common factoring works. All right, so now we have the second method in the video, the simple diamond. This is also called the x box by some teachers, but it involves this large x shape from which we'll put numbers in. I like to call it the diamond because that's what I was taught a long time ago and I'm a nostalgic person. These are used for a particular type of quadratic equation. In standard form, it would look like this. x squared, so the first coefficient, the leading coefficient is 1 plus B, which is some coefficient, times x, plus C, some constant. These both are examples of those types of trinomials and the premise of the simple diamond is to break up a quadratic trinomial into two binomial factors. Then if it's set equal to 0, you can use the zero product property to get the solution from each factor. So what we do first is when it's in this form, we set up our x box or our diamond and we say what two numbers multiply to the constant term C and add to the middle term coefficient B. We put the two here and the three here in the diamond or x box and we put the two solutions, the two numbers that satisfy both of those things in the sides here. Again, the question is what two numbers multiply to two and add to three? Well, this is a pretty easy one. You could probably guess two times one. Two times one. Now, we write the factors, the binomial factors like this. x squared plus three x plus two will equal, does equal x plus this fact, this number right here and this will be times another binomial factor, x plus one. So what we're doing is we're taking the two numbers that we determined multiply to two and add to three and we're making factors out of them. We are adding them to x in each factor, like so. Then the zero product property allows us to write each of these factors separately at equal to zero. x plus two equals zero, x plus one is equal to zero. And by subtracting the constant both sides, you could get from this one that x equals negative two, negative one. So those are your solutions to that first quadratic equation. So let's try a little bit of a harder one here. x squared plus 12x minus 45. Again, it's in this form so we can go right to the diamond. What's two numbers negative 45 and add to 12? Okay, we need to think a little bit harder about this. One of them must be negative for them to multiply together to get a negative number. They both can't be negative or they'd multiply to a positive number. So one of them's negative. That makes this a little harder. Don't think about the 12 just yet if you have a negative for the multiplication product. Think about that on its own. Negative 45. Well, nine times five. Let's try nine times negative five. Nine times negative five would be negative 45 but nine minus five is not 12. It's four. And if you switched it around so you use positive five and negative nine those would add to something negative. So that combination will not work. The next one you will think of is probably 15 times three. Let's try negative three and 15 times negative three is negative 45 and 15 minus three is 12. We have found our two numbers. Now we will make our two factors out of them. X plus 15 X minus three. You see it was a minus three here so it belongs as a minus three in the factor. And then we will set this equal to zero. Use the product property. We get X plus 15 equals zero. And X minus three equals zero. Well, we're going to subtract 15 to both sides. We're going to add three to both sides in this other equation. The two solutions we will get when we do that are that X equals negative 15 and positive three. I have one more example for you because I want you to get this. So this time we have X squared minus 11 X plus 28. It is in the proper form so the simple diamond applies assuming it can be factored which we assume in all of these so far. X squared minus 11 X plus 28. This time the term that we are going to try and get the two numbers to multiply to is positive and the term that we're going to add to if they add to a negative number but multiply to a positive number we're dealing with two negative numbers. It's the only way. So let's think about 28 14 maybe you divide it in half 14 times two. So that would be negative 14 times negative two and makes positive 28 but those don't add to 11 they add to negative 16. The next ones you might think of are seven and four. So let's do negative seven plus negative four is negative 11. Those are the two numbers. So we have X minus seven X minus four equals zero zero product property set each one equal to zero separately and solve for X that will require multiplying by those two I mean excuse me adding those two constants to both sides. So we'll get two positive solutions here seven and four so that's the simple diamond method for factoring. All right and the last type of factoring for the video the difference of squares factorization so this is done when you have quadratic equations in the following form A squared X squared minus B squared or A squared minus B squared X squared. Why all the squares? Well with difference of squares factorization you have to make sure that both terms in your binomial are perfect squares which means you can put a square root around them and you get a nice answer after you simplify that okay that's what a perfect square is so all of these are perfect squares X squared 64 9 X squared 81 144 and 25 X squared I could put a square around each of those and get a nice result not some irrational number or expression which is great that is required so you must test to make sure that they are of this form or you cannot use difference of squares also notice that there's a minus sign between the two terms if there's a plus sign between the two terms you're not using difference of squares factorization unless you want imaginary numbers of course I'm assuming this is an algebra one course or a math integrated math one course so we're not looking at imaginary numbers yet alright let's remember the premise so the premise of the difference of squares is to break the quadratic binomial into two linear binomials now you might have hopefully you've learned this already in your pre-algebra course or your algebra readiness course but a squared minus b squared is simple difference of squares where the two things could be anything as long as they're squared you get a plus b times a minus b it's the two factors that you see square-rooted with a plus in between them and one of the factors and a minus in between them and the other factor this is how difference of squares factorization works let's apply that to our first example x squared minus 64 equals zero I already said that I could square root both those things and there's a minus sign in between so immediately you could set up your two like so in the first parenthesis we will put the square root of x squared x plus the square root of 64 8 then we will place those two numbers again in the same spots in the second parenthesis with a minus sign in between and we set that equal to zero I think you could see what's going to happen based on the beginning of the video the zero-product property you would get x plus 8 equals zero x minus 8 equals zero your two solutions will be x equals negative 8 and positive 8 from that setup the next one 9x squared minus 81 yes there is a possibility of a perfect square number in front of your x squared term these again perfect square roots set up our parenthesis what's the square root of 9x squared 3x what's the square root of 81 9 first parenthesis we'll place those with a plus sign in between them the second one we'll place those two with a minus sign in between them ok if you set that equal to zero you can see it's just a little more complicated but I did want to go over it you'll get 3x plus 9 equals zero and 3x minus 9 equals zero solve for x like you would in any algebra situation we're gonna subtract 9 from both sides in this equation we're gonna add 9 to both sides in the other equation and then we will see that we have 3x equals negative 9 and we have 3x over here equals positive 9 all that's left to do is to divide both sides by 3 and you get that x equals negative 3 and positive 3 ok last one notice how the x squared term is at the end but that is just in this form and we can use the difference of squares to solve it what's the square root of 144 some of you might be thinking of a gross in a dozen that's 12 what's the square root of 25x squared it's 5x so we have 12 minus 5x equals 0 now we're gonna have to do what we did in the previous one but this time we're gonna get a fractional answer that's ok it's just I'm just alerting you to that happening so 12 plus 5x equals 0 12 minus 5x equals 0 so we're gonna start by subtracting 12 to both sides in both equations we get that 5x equals negative 12 we get that negative 5x equals negative 12 then you'll divide both sides by 5 and we see 12 fits and negative 12 fits as our two answers this one would give us positive 12 fits because these negative signs cancel each other out so our two answers here are x equals negative 12 fits and positive 12 fits ok guys that wraps up basic solving of quadratic equations by factoring next video is gonna go over some different factoring techniques that are more complicated thanks you guys for watching this is Falconator signing out