 Welcome to the third part of our four part series on solving quadratic equations. Okay, factoring. So this one's probably going to be your least favorite of all methods, but that's okay. Some of you guys are good factors, and we want to make sure we cater to everybody. So you can solve many, but not all. I mean, I like that part here. You cannot factor every equation. So unfortunately, you're not going to be able to solve all of them by factoring. But anyway, you want to set your equation equal to zero, factor it, and then set each factor equal to zero and solve. Okay, so the first one says 3x squared minus 7x plus 2. So I'm going to go ahead and set up my x here. So at the top, I need to multiply my 3 and my 2. So that's going to give me a 6. Then in the bottom part, we get negative 7. So remember, we want to multiply to 6 and add to negative 7. I believe that's going to give us negative 6, negative 1. So then I can break up my middle term with those values. 3x squared minus 6x minus 1x plus 2 equals zero. Mine's going to carry that along. Throw in our parentheses. So let's go ahead and factor by grouping. The first two have a 3x in common. So I can factor that out. I'm left with an x minus 2. The next two, now we have a minus 1x and a positive 2. So in order to get an x minus 2 left, I'm going to have to flip those signs, which means I want to factor a negative 1 out. So that leaves me with x minus 2 equals zero. Let me underline this is one term and this is another term. What did those terms have in common? I see an x minus 2 in both. And when I factor that out, there is a 3x minus 1 left. Okay, now that I've gotten them factored, I can set my factors equal to zero. So I end up with x equals 2 as one of my answers. And add one of both sides on this one. Divide by 3. So x equals 1 third for my other answer here. Now if I had been doing this one with my calculator, I would need to, two would have shown up on my table. One third would not have though. So I would have had to do one with a zero and one, well you could have done this one with a zero as well. You could have eyeballed them as two. But getting exactly x equals 1 third is not going to be as easy on your calculator. Okay, fantastic. Next one, 5r squared minus 6 equals 29r. Okay, remember our first rule with these is that we need to make sure we set our equation equal to zero. So I need to move my 29r over. And just to make sure I put it in the right, let's see, the right form, I'm going to go ahead and stick that in the middle. Because remember the order of this stuff doesn't matter as long as you keep the signs that you need with it. So I'm going to write this as 5r squared minus 29r minus 6 equals zero. Okay, again we need to multiply our a and our c together. So at the top of my x, I'm going to put a negative 30. That's my multiply number. The bottom of my x, I'm going to put a negative 29. That's my adding number. So then numbers that work here are going to be negative 30 and positive 1. Okay, so let me go ahead and break those apart then with my middle term. 5r squared, oh, and I already broke my rule here. Oh, no I didn't. Okay, we want to put the negative one first. So I guess I did remember my rule. Minus 30r plus 1r minus 6, remember it was our constant term, equals zero. Okay, throw in my parentheses. Let's factor this by grouping so I can take a 5r out of the first two terms and that leaves me with an r minus 6. I already pretty much have an r minus 6, so I can factor 1 out, r minus 6, so that gives me a zero. Looking at this is one term, this is another term. I can factor my r minus 6 out and that leaves me with 5r plus 1 equals zero. Okay, set it equal to zero, factored it. Now I need to set my factors equal to zero. So r minus 6 equals zero, 5r plus 1 equals zero. So for this one we can add 6, r equals 6, that's one of my answers, this one I could subtract 1, 5r equals and negative 1 divided by 5, sorry I'm at the bottom here, so r is going to equal negative 1 over 5. So again, one of these would have been pretty nice to find on the calculator, the other one, not so much. Okay, now for everybody's favorite, the good old quadratic formula. Okay, and if you have a time, Google this and see if you can find out the song that goes along with this equation, it's quite snappy, but I'm not going to sing it for you. Trust me, you'll appreciate that. So you want to manipulate the equation until it's in the form, let me go and highlight that form, ax squared plus bx plus c equals zero. So our typical standard form for a quadratic, but you want to make sure again it equals zero. Identify your a, which is the coefficient of your x squared, your b, the coefficient of your x, and your c, the constant term. Plug these values into the quadratic formula, which is x equals negative b plus or minus the square root b squared minus 4 times a times c all over 2a. Okay, and simplify, and that's going to be the hard part. Okay, first equation says 6p squared plus 15p equals 21. So you notice I used a standard ax here in my equation in the formula, so since we have a p as our variable, we're going to have to write p equals, but that's an easy thing to fix. Okay, we have to set this thing equal to zero, so that means I need to move my 21 over, so minus 21, minus 21, so my standard form is going to look like 6p squared plus 15p minus 21 equals zero. So identifying my a is 6, my b is 15, and my c is negative 21. Okay, don't forget your signs too, because if you lose this negative, we're going to get a totally different answer. All right, so the quadratic formula p equals the opposite of b, so negative 15 plus or minus the square root of, so we need 15 squared minus 4 times 6 times negative 21, and all of this is over 2 times my a value, which is 6. Okay, now we get to simplify this. So let's go to the calculator. I'm going to plug in everything that's underneath the square root into my calculator, and then let's see if we get a nice number or not. So 15 squared, I'm going to put that in parentheses, although for this particular problem, it's not really necessary, minus parentheses 4 times 6 times negative 21. All right, let's see if we get a nice number here. That gives us an answer of 729. I believe that is a perfect square. So we get negative 15 plus or minus the square root of 729, all over 2 times 6, which is 12. Okay, that's going to equal, so let's do the square root of 729, unless you know that one off the top of your head. Square root 729 gives me an answer of 27. So this is a negative 15 plus or minus 27, all over 12. Now, for me, these are perfectly good answers, but they're kind of goofy looking, so let's go to the calculator and see how we can actually get numbers that look more like numbers out of this. So we want to do, in parentheses, negative 15 plus 27 and then divide that by 12. So that gives me 1, oh, that's a very nice number. And then we can also do parentheses negative 15 minus 27, all divided by 12. All right, that gives us negative 3.5. So my answers are, if you want to list them out separately, p equals 1, p equals negative 3.5. So we could have graphed this one. These would have shown up fairly nicely on the graph. We could have probably factored this one as well. So again, I'm just giving you lots of options that you can do with these problems, just to make sure, you know, you know how to do them and you can use a variety of ways to do it. All right, last one here wants us to solve the equation x times the quantity 4x plus 3 equals 10. So remember, I need this in standard form, so I'm going to have to do some manipulation here, which means I'm going to need to multiply this x through both sets of these parentheses and I'm going to have to move this 10 over to the other side. Okay, so that's going to look like x 4x squared when I multiply that x through, plus 3x minus 10 equals 0. So again, I multiplied this out and I subtracted 10 from both sides in order to get this part of the equation. All right, let's use a is 4, b is 3, and c is negative 10. Okay, so this time x is our variable, so x equals the opposite of 3 plus or minus the square root of 3 squared minus 4 times 4 times negative 10 all over 2 times 4, beautiful. So again, I'm going to go to my calculator. Let's put in all that stuff that's under that square root. So 3 squared minus parentheses 4 times 4 times negative 10 and that gives me a grand total of 169 under that square root. So that's negative 3 plus or minus the square root of 169, another perfect square, all divided by 8. And let's say this number hadn't been a perfect square, then for me you can just leave it. If you're doing this in one, an actual estimation though, you're going to have some ugly decimals for your answer. Okay, so we're going to end up with x equals, I believe the square root of 169 is 13, but let's double check. My mental math sometimes fails me. Okay, 13, fantastic. So we have negative 3 plus or minus 13 all over 8. So again, that would be a perfectly legitimate answer, but let's go ahead and figure out what these are then separately so we can break them up. So that's going to be negative 3 plus 13, which is going to give us 10 and then divide that by 8. So we get an answer of 1.25. Then we have negative 3 minus 13, which is negative 16 divided by 8, so our answer should be negative 2. So x is going to equal 1.25 or 5 fourths if you want to write it like that. And x is going to equal negative 2. So those are my two answers. Right, fabulous. Keep practicing these and we'll meet again next time.