Japanese| A very interesting Math Olympiad algebraic problem| Solve for a and b

Hi, you're welcome. Let's solve for the values of a and b in this particular equation with the given condition. that a is not equal to b, stick around and trust me—you're going to enjoy this, right? Okay, this Let's call this Equation One and this Equation Two. Okay, so if we say Equation One minus Equation two: so that will be a squared minus b minus b squared minus a, right? Is equal to 73 minus 73 so we have a squared minus b minus times b squared gives us minus b squared minus times minus a plus a equals zero, okay, so we can factor out a squared minus b squared from here, then plus a minus b equals zero; now, a squared minus b squared is equal to a minus b multiplied by a plus b. All right, now plus a minus what b is equal to zero. Now, if you look very closely, you'll notice that we have a minus b, a minus b, so let's factor it out. So if we do that, we are going to have when the the whole of this divided by this leaves us with a plus b, and when it divides this, we have Plus one equals zero, all right? There's no need for this. There's no need for this, yeah. so zero, so that means we can say that a minus b is equal to zero, okay? But in this case, a is equal to b and record that we are told that a is not equal to b as one of the conditions, so we're not I'm going to make use of this, all right, right now. The second one is that a plus b plus one equals zero. That means a plus b equals minus one. Let's call this Equation Three, a very important equation. We're getting back to it. We're going to come back to it. All right, now let us—we've added a Subtract equation two from equation one. Let's add both of them together. So if we say equation One plus equation two, so what we are going to have is a squared minus b okay plus b squared. Now let's solve so we can factor out a squared plus b squared from here, then we have minus a. minus b equals what? 146. Okay, so we have a squared plus b squared equals 146, then if This becomes plus a, the same thing b becomes minus b becomes plus b, so we have a. a squared plus b squared equals 146 minus one; yes, record that a plus b according to the equation three is minus one, okay, so that means that a squared plus b squared is equal to 145. Now let's Call this equation (4). From (3), we have that a plus b is equal to minus one. if we square both sides of the equation, we are going to have a plus b squared is equal to minus one squared, right? So a plus b squared simply means a squared plus two a b plus b squared. is equal to one, okay, because minus one squared is equal to one. Now let's continue from here. So we can bring out a squared plus b squared here, right? So we have a squared plus b squared. plus two ab plus two ab equals one; we have the value for a squared plus b squared, and that is What is 145? So we have 145 plus 2ab equals 1. If we make 2ab the subject of the formula. We are going to have one minus 145, right? One minus 145. Okay, so we have two a b is equal to minus 144, so our ab will be minus 144 divided by two, so our ab will be equal to minus 72 now this is another important thing, let's call it equation five, okay? Now don't worry, very soon You are going to see why we are getting all these equations; we are going to substitute to get our a. And our b. All right, so let us find a minus b squared according to the algebraic identity is Alright, so a minus b squared is equal to a squared plus b squared minus two ab, so that. means that a minus b squared is equal to a squared plus b squared, which is what, um, 145, so We have 145 minus two times the product of a and b. Recall that a times b is minus 72, so we have a minus b squared. is equal to 145 minus plus, because minus minus is plus, so we are going to have 144 here, right? 144 Okay, let me clean this side because we don't need anything here except equation three. And equation three states that a plus b equals minus one, so that is the only thing I'm going to leave. We're going to need it, so apart from that, all these others are useless, okay? So this is what We have that a minus b squared is equal to 145 plus 144. Let's calculate: plus 144 will be 289, right? 289 All right, so 145 plus 144 will give us 289, so our a minus b should be equal to the square root of 289, so that means a minus b will be plus or minus 17. Okay, um, now let's call this equation. What six? Yes, if this is the first time you're coming across this channel, please don't hesitate. to subscribe and turn on your notification bell so that you'll be notified any time I upload A new video. Okay, so this is equation six. If we should compare equation six and equation three. because equation three states that a plus b equals minus one, and we have obtained that a minus b equals equal to plus or minus 11, meaning that there are two cases here. Now, what are the cases? We have case One case here is that's what a minus b is equal to 17, and case two implies that a minus b is equal to minus 17, so we are going to consider these two cases. All right, so let's start with case. One for case one, okay. Comparing this equation, six comparing six and three. All right, so we are going Choose case one, which is where a minus b equals 17, so we have a minus b equals 17. And equation three states that a plus b equals what minus one. Now let's solve both of them. simultaneously because we have two different signs here, we are going to add so that we can eliminate our b, okay? So if we do that, a plus a will give us 2a, then minus b plus b will give us zero. zero is equal to 17 plus or minus one, which gives us 16, so from here we can say that a is equal to So let's call this a one. All right, now let's compare this for case two, for case two. Let's find a pair for case two: it will be what a plus a minus b equals minus 17. This is our Case two: that's a minus b equals minus 17, okay, and a plus b according to our equation three. is equal to minus one, so if we subtract as well, sorry, if we add a plus a, we get 2a, all right. is equal to minus 17 minus 1, which gives us minus 18. Now, if we solve this, let me clean this up—I think We have obtained what we wanted, so if we solve this, we can see that our a2 is going to be negative. Nine, all right. So we have found our A1, and you have found our A2. Now let's go ahead and get our B1 and B2 respectively, so for this particular case, that is for case one, to get our b2 from here to get our b1, sorry, we notice that our a here is eight, so let's substitute this eight. into any of these equations to get our b1, so if we substitute into this first equation, we can say that a which is 8 minus b is equal to 17. Recall we are looking for b1; we have obtained a1, and we are Looking for the corresponding b, okay, so that means that minus b1 is equal to 17 minus 8. So minus b1 equals 17 minus 8, which gives you 9, right? We give you 9, so b1 becomes minus 9. That means a1 is equal to 8 and b1 is equal to what? Let's write it somewhere: a1 comma b1 is equal to 8, minus 9. Perfect. Now let's go to the second case. For the second case This case is all right. Let's substitute b; our a is minus 9. Let's substitute it into any of this. equation to get our b2, okay, so if we substitute it into the first equation, we are going to have a is minus 9, right? Minus 9 minus b is equal to minus 17. Okay, now minus 9 plus 17 is equal to Positive now 17 minus 9 gives us 8, right? So we have that 8 is equal to b. So that is our B2; that means A2 comma B2 is equal to. Now, what is our A2 minus 9? And what is our B2-8 now? Let's check. Let's check whether what we solved is correct. So we're going to use The first equation: a squared minus b is equal to 73. a squared minus b is equal to 73. All right. Now, um, let's use our first values for a and b. So a is what, 8? So we have a squared minus b is minus. 9 minus 9 equals 73, so 8 squared is 64 minus minus is plus, which equals 73, so 73 equals 73. You can use this one to check for any of the you can check this into any of these equations To confirm your solution, thank you so much for watching, have a lovely day.