Italian| A Nice Math Olympiad exponential problem|Solve for x.

Hi, when I saw this question, I was like, no, no, no—I must share the solution to this particular question. with my lovely subscribers, okay, now we are asked to solve for this: if x squared minus x plus one is equal to zero, all right. This is how to go about it. First, we are going to concentrate on this. and bring out some vital equations from there so we have x squared minus x plus one equals Okay, so that means our x squared is equal to if this minus x crosses it, it becomes plus x. If this plus one crosses, it becomes minus one, so we have that x squared is equal to x minus one. Now let's also call this Equation One. From this main equation, if we say x squared minus x, let's Transfer this one to the other side, it becomes minus one, right? So we have x in common, this and This has x, we have x in common, so we bring out x, then we open our bracket: x squared divided by x. We have x minus x divided by x, we have one is equal to minus one. Okay, now let's divide both sides. to this equation by x; when we do that, you will notice that your x minus one over x minus one is equal to minus one over x. Now let's call this equation two. All right, can you see something about this equation? One is equal to x minus one; equation two is also equal to x minus one, so we can actually say If we compare equation one and equation two, then x squared is equal to minus one over x. so we are going to write that x squared is equal to minus one over x, okay? Now what if we Decide to multiply both sides of this equation by x so as to eliminate this denominator. so we have x squared multiplied by x equal to minus one over x multiplied by x So this will cancel out; we have x squared times x, which gives us x to the power of three. Yes, since a to the power of y multiplied by a to the power of x is equal to a to the power of y plus x. plus x with the same base, you add the exponents, okay? In this case, the exponent of our x is one, all right, so we Let x cubed be equal to minus one. Now, let's call this Equation (3), okay. right if this is the first time you're coming across this particular page or this channel вероятно, вы хотите, чтобы я перевел этот текст на английский язык. вот перевод: Whenever I upload a new video, all right, okay, now, as I said, the solution to this is truly amazing. Stick with me to the end and you're going to be glad you did. Now the question says x to the power of 2020 plus x to the power of 10, 10 minus one is equal to what? So let's start with x to the power to manipulate it, okay, now x to the power of 2020 can be written as x to the power of one multiplied by x to the power of 2019, recall what I said: when you have the same base, you add the exponents, so when You add this, you are going to get this back. Now, this 2019 can be written as x to the power of 2019. can be written as x raised to the power of 3 multiplied by so 2019 divided by 3 let's see what we have 3 we divide 20 to give us 6, this remainder 2; if we divide 21 we have what 7; if we divide 9 we have 3, so we have 6, 7, 3, okay. All right, so we are now going to replace this x to the power of 2019 with x to the power of 3 times brackets 6, uh, 3 multiplied by 6, 7, 3, so we can now rewrite this x to the power of 2020 is equal to x raised to the power of 1 multiplied by x raised to the power of 3, all within brackets, 6, 7, 3, okay, can be x to the power of 2020 is equal to x to the power of 1 multiplied by x to the power of 3 raised to the power of 6, 7, 3, right? So we have x to the power of 2020 equal to x to the power of 1. multiplied by x to the power of 3 is minus 1, recall, right? Yes, isn't that amazing? So we have minus 1 raised to the power of 673, okay, so we have that x raised to the power of 2020 is equal to x to the power of 1, which is still x, multiplied by minus 1 since this is odd, so minus 1 raised to the power of an odd number will still give us minus 1, so we have that x raised to the power of 2020 is equal to minus x, so let's call this equation 4. All right, now let us manipulate x to the power of 10, 10 x to the power of 10, 10. This is sent in as x to the power of 2. multiplied by x to the power of what? 1008. Now, why are we choosing this? I chose x to the power of 2. because I know that if I choose x to the power of 1, what I will have here is 1009, and this cannot be divided by 3 because we know you, we have x to the power of 3, we have a value for it, so that's why x the power of 2 here so that I'll get a number that I can multiply by 3 this way Okay, so we have x to the power of 10, which is equal to x to the power of 2 multiplied by then we have here to be x to the power of let's divide this by 3 and see what we have Divide that by 3. 3 divides itself. When we divide 10, we have 3 with a remainder of 1. You divide 10 again, 33 remains; that one, if you divide 18, 6, so this will be x to the power of 3. multiplied by 3, 3, 6, okay, all right, now let's solve this, implies that x to the power of 10, 10 is equal to x squared multiplied by recall that this is the same as x cubed raised to the power of 3, 3, 6, okay, so we continue, we continue, we continue, we continue So let me write what we have here: we have x to the power of 10, 10, right, is equal to x squared multiplied by x cubed, all raised to the power of 3, 3, 6, all right Now we can write x to the power of 10, 10 is equal to x to the power of 2 multiplied by this is minus 1 right raise the power of 3 to the 3rd power, 6, but since we have an even number here, that means we are going to have plus 1 here, so we have that x to the power of 10 plus 1 is equal to x to the power of 2 multiplied By 1, that means that x to the power of 10 to the 10th is equal to x squared, so this is another important point. Words, we are close to the solution, yes, we are. Okay, now to the question that we are asked to. solve x to the power of 2020 plus x to the power of 1010 minus 1 equals what? Right now we have solved it. x to the power of 2020 and we got minus x, so we can write minus x plus that we solved x to the power of 10 10 and we have x squared plus what x squared minus 1 equals what all right now let's Let's rearrange this. We write x squared minus x minus 1 is equal to what, right? Okay, now from this first equation we have that x squared minus x plus 1 equals 0. That means x squared minus x equals this minus 1, because the plus 1 crosses over to become minus 1. Alright, so we can now replace x squared minus x here with minus 1. So if we do that We'll have that minus 1 minus 1 is equal to what minus 2, and this becomes our solution, isn't that? That's very, very lovely. Thank you for watching. Have a bright, lovely, and beautiful day. Bye.