CA Algebra I: Quadratic equation

We're on problem 53. It says Toni is solving this equation by completing the square. ax squared plus bx plus c is equal to 0, where a is greater than 0. So this is just a traditional quadratic right here. And let's see what they did. First, he subtracted c from both sides and he got ax squared plus bx is equal to minus c. OK, that's fair enough. And then let's see. He divided both sides by a. Right, that's fair enough. He got minus c/a. Which step should be Step 3 in the solution? So he's completing the square. So essentially, he wants this to become a perfect square. So let's see how we can do that. So we have x squared plus b/a x-- and I'm going to leave a little space here-- is equal to minus c/a. So for this to be a perfect square we have to add something here, we have to add a number. And we learned from several videos in the past and we kind of pseudo-proved it. And actually, I have several videos I do solely on completing the square. You essentially have to add whatever number this is, add half of it squared. And if that doesn't make sense to you, watch the Khan Academy video on completing the square. But what is half of b/a? Well it's b over 2a. So 1/2 times b/a is equal to b over 2a. And then, we want to add this squared. So let's add that to both sides of this equation. So we're left with x squared plus b/a x. And we want to add this squared. Plus b over 2a squared is equal to minus c/a. Anything you add to one side of the equation, you have to add to the other. So we have to add that to both sides. Plus b over 2a squared. And let's see if we've solved the problem so far, what they want. X, b over 2-- right. This is exactly what we did. x squared plus b/a plus b over 2a squared, and they add it to both sides of the equation. So D is the right answer. Now if you find that a little confusing or if it wasn't intuitive for you, I don't want you to memorize the steps. Watch the Khan Academy video on completing the square. Next problem, 56. No, 54. All right, this is another one that should be cut and pasted. All right, four steps to derive the quadratic formula are shown below. I said in previous videos that you can derive the quadratic formula by completing the square. And we actually do that in another video. I don't want to give too much of a plug for other videos, but let's see what they want to do. What is the correct order of these steps? So the first thing you want to start off with is just a quadratic equation. And this one is the first step. This is where we started off with in the last problem. Then what you want to do is add 1/2 of this squared to both sides. So b over 2a squared you want to add to both sides, and that's what they did here. So our order is I. And then you want to do IV. That's what we did in the last problem. We did IV. And then from here, you know that this expression right here is going to be equal to x plus b over 2a squared. And once again, watch soon. the completing the squared video if that didn't make sense. But the whole reason why you added this here is so that you know that, OK, what two numbers, when I multiply them equal b over 2a squared, and when I add them equal b/a? Well that's obviously, b over 2a. If you add it twice you're going to get b over a. If you square it, you're going to get this whole expression. So you say, oh, this is just x plus b over 2a squared and you get that there. And then, is equal to-- and then they just simplify this fraction. They found a common denominator and all the rest. And so the next step is Step II. And then all you have left is Step III. And you've pretty much derived the quadratic equation. So I, IV, II, III. That's choice A. Problem 55. Which of the solutions-- OK, I'll put all of the choices down. So which is one of the solutions to the equation? So immediately when you see all of the choices, they have these square roots and all that. This isn't something that you would factor. You would use a quadratic equation here. So let's do that. So the quadratic equation is, so if this is Ax squared plus Bx plus C is equal to 0. The quadratic equation is minus b. Well they do it lowercase. Plus or minus the square root of b squared minus 4ac, all of that over 2a. And this is just derived from completing the square with this, but we do that in another video. And so let's substitute it in. What is b? b is minus 1, right? So minus minus 1, that's a positive 1. Plus or minus the square root of b squared. Minus 1 squared is 1. Minus 4 times a. a is 2. Times 2. Times c. c is minus 4. So times minus 4. All of that over 2a. a is 2, so 2 times a is 4. So that becomes 1 plus or minus the square root. So we have a 1. So we have minus 4 times a 2 times a minus 4. That's the same thing as a plus 4 times 2 times a plus 4. Let's just take that minus out. So it's plus. There's no minus here. So let's see, 4 times 2 is 8. Times 4 is 32. Plus 1 is 33. All of that over 4. Let's see, we're not quite there yet. Well they say, which is one of the solutions to the equation? So let's see. If we wanted to simplify this out a-- well, this is right here. Because we have 1 plus or minus the square root of 33 over 4. Well they wrote just one of them. They wrote just the plus. So C is one of the solutions. The other one would have been if you had a minus sign here. Anyway, next problem. 56. And this is another one I need to cut and paste. It says, which statement best explains why there's no real solution to the quadratic equation? OK, so I already have a guess of why this won't have a solution. But in general-- well, let's try the quadratic equation. Before even looking at this problem, let's get an intuition. It's negative b plus or minus you the square root of b squared minus 4ac, all of that over 2a. My question is to you, when does this not make any sense? Well you know, this'll work for any b, any 2a. But when does the square root sign really fall apart, at least when we're dealing with real numbers, and that's a clue? Well, it's when you have a negative number under here. If you end up with a negative number under the square root sign, at least if we haven't learned imaginary numbers yet, you don't know what to do. There's no real solution to the quadratic equation. So if b squared minus 4ac is less than 0, you're in trouble. There's no real solution. You can't take a square root of a negative sign if you're doing with real numbers. So that's probably going to be the problem here. So let's see what b squared minus 4ac is. You have b is 1. So 1 minus 4 times a. a is 2. 2 times c is 7. And sure enough, 1 times 4 times 2 times 7 is going to be less than 0. So let's just see what they have here. Right, the value of 1 squared-- oh, right. It's b squared. Well 1 squared, same thing as 1. 1 squared minus 4 times 2 times 7, sure enough is negative. So that's why we don't have a real solution to this equation. Next problem. I'm actually out of space. OK, they want to know the solution set to this quadratic equation. I'll just copy and paste. So that's essentially the set of the x's that satisfy this equation. And obviously, for any x that you put in this, the left-hand side is going to be equal to 0. So what x's are valid? And they just want us to apply the quadratic equation. So we've written it a couple of times, but let's just do it straight up. So it's negative b. b is 2. So it's negative 2 plus or minus the square root of b squared. Well that's 2 squared. Minus 4 times a. a is 8. Times c, which is 1. All of that over 2 times a. So 2 times 8, which is equal to minus 2 plus or minus the square root of 4-- let's see. Did I write this down? Negative b plus or minus the square root of b squared minus 4 times a times c. Right. So you get 4 minus 32. That's why I was double checking to see if I did this right because I'm going to get a negative number here. All of that over 16. And so we're going to end up with the same conundrum we had in the last. 4 minus 32, we're going to end with minus 2 plus or minus the square root of minus 28 over 16. And if we're dealing with real numbers, I mean there's no real solution here. And at first I was worried. I thought I made a careless mistake or there was an error in the problem. But then I look at the choices. They have choice D. And I'll copy and paste choice D here. Choice D. No real solution. So that's the answer, because you can't take a square root of a negative number and stay in the set of real numbers. Let's see, do I have time for another one? I'm over the 10 minutes. I'll wait for the next video. See