 Welcome to the last video on quadratic equations. So we are going to be solving this one and we're just going to mix it up as I call it here. So some quadratics can be solved by factoring, okay, but not all of them unfortunately. Some can be solved using the square root property. Okay, so that's another option. Next option, all quadratics can be solved by graphing unless it doesn't have a solution. And then the fourth option here is the quadratic formula. So again, all quadratics can be solved by setting your equation equal to zero and using your quadratic formula. Not all quadratics though have real solutions, so you may come across some of them where you'll end up if you're doing the quadratic formula where you have a negative under the square root that can't happen or if you're graphing it, you'll end up either with two things that don't cross or something that doesn't have any x intercepts. So just be aware that there are some out there that, you know, you can't do, you can't solve every equation unfortunately. Okay, so we're just going to look at each problem that I have here and we're just going to pick a method that we think might be good and we'll give it a shot. So first equation is 3x squared minus 7x plus 2 equals zero. This one I believe can be factored. That makes sense. Using the square root property though, does it make any sense because I see an x squared here but I also see an x. So unfortunately with that method, remember you only want to have one square. So that one doesn't make sense here. We could graph it or we could use the quadratic formula. Okay, so since I have to pick one of them, I'm going to start with factoring. Let's start at the top. So I'm going to make my x because this one has a coefficient in front. I'm going to multiply my a which is 3 by my c which is 2 and that's going to give me a total of 6. My number at the bottom then is negative 7. So that means I need two numbers that multiply to 6 and add to negative 7. So I believe in this case that will work. Negative 6, negative 1. Those two multiply to 6, add to negative 7. So let me go ahead and break up my middle term. So that's going to look like 3x squared minus 6x minus 1x plus 2 equals 0. Okay, I'm going to throw my parentheses in to factor by grouping and I can factor a 3x out of my first set. That leaves me with x minus 2. And looking at my second set, I have a negative x and a positive 2 so that means I want to factor a negative 1 to get my signs to work out correctly. So that's going to leave me with x minus 2. And when I look at this is one term, this is another term, that's going to mean I have an x minus 2 in both of them so I can factor that out and I'm left with a 3x minus 1 equals 0. Okay, set each factor equal to 0 and we end up with x equals 2 and x equals 1 third. And if you need help getting to that point, refer back to one of the previous videos that we've done to show you how to set factors equal to 0. Okay, so factor and work with that one, fabulous. Okay, let's look at the next one. So n squared minus 2n minus 3 equals 0. So for this one, we could probably factor this. I think that'll factor nicely. We can't use the square root property again because I see an n squared and an n so that doesn't make sense here. Let's go ahead and graph this one since we haven't done a graphing one yet. So go to our y equals, obviously we're not going to use an n for our variable so we're going to use x squared but we have to remember then to translate that properly when we go to write our answer down. Minus 2x minus 3, let's check our table and see what pops up here. Sure enough, I see a negative 1 and because my signs are both negative, that means one's going to be positive and one's going to be negative. So let me scroll this direction and sure enough, my other one's 3. So that means I could have factored this. Let's go back to that idea. This would have factored as, let's write down our solutions, n equals negative 1 and n equals a positive 3. So going backwards, that means this would have factored as n plus 1 times n minus 3 equals 0. When you set it to those factors equal to 0, you would have got these two answers. Okay, interesting. Okay, number 3, this one says k squared plus 6k equals negative 5. So again, using the square root property doesn't make sense because I have a k squared and a k. So this one we'll use the quadratic formula on, since we haven't done that one yet. So that means I need to set this equal to 0, k squared plus 6k plus 5, moving that negative 5 over equals 0. So identifying my coefficients, a is 1, b is 6, my panel cooperate here, 6, and c is 5. So remember the quadratic formula is k equals the opposite of b, so that's going to be a negative 6 plus and minus the square root of b squared. So that's 6 squared minus 4 times a, which is 1, times c, which is 5, all over 2a, which is 1. Okay, let's go to the calculator and figure out what this number is inside of that square root. Put it out of here, so we're going to do 6 squared minus 4 times 1 times 5. I got a little lackadaisical with my parentheses, too, but when it's positive, it's not going to matter. Okay, that gives me a 16, so that's going to give me negative 6 plus or minus the square root of 16 all over 2. Square root of 16 is better known as 4, so that is negative 6 plus or minus 4 all over 2. So negative 6 plus 4 gives me negative 2 over 2, so k is going to equal on negative 1, and negative 6 minus 4 all over 2 is going to give me a negative 5, so those would have been my two answers for k. And again, you don't have to do this the way I'm doing it, but you shouldn't have put the same answer that I get. Okay, number 4, let's erase some of this mess that I made here. So this one says negative 2 times the quantity of x plus 4 squared equals negative 28. So finally, because I only see 1 square, it's a squared quantity, woo, we can use our square root property. Okay, so let's do that on this one. Go ahead and remember we have to isolate that square. So divide both sides by negative 2. So that's going to leave me with x plus 4 quantity squared. Oops, why did I do that? Try that again. My square went on the wrong place. x plus 4 quantity squared equals, that's going to give us 14. Okay, then remember to undo the square, you want to apply the square root, and don't forget about the symmetry idea, so there's a plus and a minus in front. So that's going to leave me with x plus 4 equals plus and minus the square root of 14. So I'm going to go ahead and subtract 4 from both sides. So x equals negative 4 plus or minus the square root of 14. Now on this one, I will tell you, if you decided to factor this, you would have to multiply this whole stuff out, move stuff around. Same thing with the quadratic formula. So in this case, using the square root property was a lot quicker. And also, you notice I get the square root of 14. That's not a very pretty number. Okay, next one, let's go ahead and graph this one again, just to give us something different to do. So let's go to the calculator and graph. We'll do the intersection this time as well, since there's two sets of part equals sign. So 2x squared plus 7x equals negative 3. So in my y1, I've got my first side of my equation. In my y2, I'm going to put minus 3. Okay, let's graph this on our standard window. There's my parabola. There is my line. Okay, do you think I can get lucky and find values on the table? Looks like both of my answers should be negative. So let's take a look in the negative area. Oh, looks like we got lucky with one of them anyway. And I think that's all we're going to get. So you notice that negative 3, the y values match, and they're both negative 3. So we need to find the other one then by going to the intersect. So second calc, option number 5. So negative 3 must be that one. So we need to find the one over here to the right. So I'm going to toggle to the left though, since they're both on the negative side. And that's pretty darn close. Hit enter, hit enter, hit enter. And we get an answer of negative 5. Okay, so let me go ahead and draw my graph on here. Here's my lovely parabola. Oops, put my axes in too, sorry. Here is negative 3. So my intersections are here and here. So those were at negative 3, negative 3, and negative 0.5, negative 3. So my axes for this one are negative 3 and negative 0.5. Okay, next one. We have 5r squared minus 6 equals 29r. Okay, we haven't factored in a minute. So let's see if this one will factor for us. So we're going to have to move this 29r over and set this equal to 0. No, actually, we haven't done the quadratic formula yet. So let's do that one instead. I changed my mind. Okay, so that's going to give me 5r squared. Now to put it in standard form, I'm going to do minus 29r next, minus 6 equals 0. So my a is 5, my b is negative 29, and my c is negative 6. So r is going to equal the opposite of negative 29, so that's going to give me 29 plus or minus the square root of negative 29 squared, and this one, we definitely better use our parentheses on when we put in the calculator. Minus 4 times 5 times negative 6, all over 2 times 5. Okay, let's go to the calculator. Quit out of my graph. So I'm going to put parentheses negative 29 parentheses squared minus parentheses 4 times 5 times negative 6, parentheses, hit enter. That gives me a grand total of 961, okay. So 29 plus or minus the square root of 961 all over 10, when I multiply 2 by 5. Let's see if 961 is a perfect square, or if we're just going to leave it. Okay, it sure is. It gives us 31, so my answer is going to be 29 plus or minus 31 all over 10. And I'm going to leave my answer just like that. It's perfectly legitimate. Okay, last problem, 7x equals 6 minus 20x squared. Okay, we just did graphing, we just did factoring. We've pretty much done all of our methods here now today. So I think graphing is the easiest. I'm going to go back to that. Go to my y equals, clear both of these out. So I have 7x, my y1, I'm going to do 6 minus 20x squared in my y2. Hit graph, there's my line, there's my parabola. Okay, it looks like one answer is positive, one answer is negative. Let's look at our table and see if we get lucky. So we want a, we don't want anything nice here. So we want these two values to be the same, which it doesn't look like they are anywhere in here. Let's see in our positive area, nope. Okay, so we definitely better use the intersection on this one. So second calc, option number five, intersect. I'm going to go to the right first, hit enter, hit enter, hit enter. I get 0.4, okay, do it again, second calc, option number five. Let me toggle to the left this time. It's pretty darn close, hit enter, hit enter, hit enter. And we get negative 0.75. Okay, so we draw on a rough sketch on my graph here then. Ooh, that's pretty bad, but that's okay. All right, then they cross here. And I didn't remember what the y values are, but that's okay. We just need our x's. So x is going to equal negative 0.75 and x is going to equal 0.4. Okay, fabulous, we're done with quadratic equations. Yay, keep practicing these and we'll be back later.