A Math Olympiad algebraic question|solve for x

We have an interesting question on the board. Let's find the value of x, and x must be real. Okay, all right, let's go. First of all, this is the question: x to the power of 27. plus x to the power of 18 is equal to 36, all right. Let's try to make this a polynomial. Okay, and how do we do that? We need to factor out something that we can raise to the power. three plus that same thing raised to the power of two will give us 36. Now when you look at You will notice that if you take this, 27 can be written as nine multiplied by three. Right, and 18 can be written as the same nine multiplied by two, so let's go ahead and do that. Here, so we have x to the power of nine raised to the power, okay, not yet x to the power of nine. multiplied by three, okay, plus x to the power of nine multiplied by two is equal to what 36 now recall the law of indices that states that x raised to the power of m raised to the power of n is equal to x raised to the power of m times n of what n is the same thing as x raised to the power of m times n, that is m times n, just like we have It's like this: so if we want to transform it into this form, we can actually bring in one of those. exponents inside our bracket, right? So let's do that. So if we should do that, let's bring in what They have in common nine and nine. All right, so we can write x to the power of nine raised to the The power of three plus x to the ninth power squared equals 36. There you have it. Yes, now we can now form a cubic polynomial very easily, right? So now say x to the power of nine. be equal to what we'll call y, so once we're able to do this, we can now plug it into this equation and we have y cubed plus y squared equals 36, tada here it is the Almighty Polynomial we've been looking for, so this is the route that we are going to take that will lead us to the value of our x, so we go ahead and solve this particular equation There are several ways we can use to solve this. The first method is this, depending on the length. of the video I might introduce the second method, but if the video seems to be too long, I'll just Cut it at the first method, so just pray the video isn't too long, okay? Let's continue now when you Look at this; we can actually bring this in, and then you try an error to get the value of y. Then use the synthetic division method to obtain a quadratic equation, okay—that is the first method. Let's do it first. Now we have y cubed plus y squared equals 36. Okay, so like I said, letdataArray in, this 36, let's bring it in, so we have y to the power of three plus y. squared minus 36 equals zero; yes, this is plus 36 if it crosses, then the equation becomes minus 36 Now at this point, as I said, we are going to start plugging in values for y that will make this Is this statement true if we say that y is equal to one? Is this equation true? so let's check the white one to the power of three plus one squared minus 36—definitely this is not equal to zero, so definitely our y cannot be one. Now let's check two: if we say two to the power three plus two to the power of two minus 36; this is also not equal to zero, okay, no problem Now let's check three. Let's see if three will work: three to the power of three plus three to two to the power of two minus 36, yes, this is zero because this is 27 and this is nine, and 27 plus nine gives us 36 and 36 minus 36 is definitely zero. Now we have what we want. So how do we now? Make use of this finding; this simply means that one of the words, one of the roots of this The equation is y equals three, but we don't need it in this form; we need it as a factor now. As a factor, we need to set it equal to zero, and to do that, we need to subtract what? subtract three from both sides of the equation to make this place zero, so we are going to have that y minus three equals three minus three, okay, so we have y minus three equals zero, so this is Sorry, this is a factor right now; now we have y minus three equal to zero as a factor. so let's write it: y minus three is equal to zero. Now recall that we substituted y to be equal to zero here, and we get this: y equals three. Sorry, so recall that our y is equal to three. Now this is how we use synthetic division to obtain our quadratic equation. So we have these three, okay, and this is our equation. So we are going to bring out all the Now, the coefficient of y cubed in this particular equation is one. the coefficient of y squared is one; the coefficient of y is supposed to be y here, which is not there is zero, then we have minus 36. Okay, then we are going to bring down this one. Okay, so we are going To multiply three by one three times, we get three, so we add it here; we add one plus three. We give us four, we also go again, so three times four we give us 12, we add zero plus 12 we give. So okay now three times 12 gives us 36, we add again, and when you add this time you get zero, so this forms what the quadratic equation that we are going to multiply By this, give it back to us, give us this one back, okay? Now, how do we get the quadratic equation? From here, it's very simple; if this means that this is y squared, since it's quadratic. It starts from power two, so y to the power of two, that is one y squared plus four y. Right plus 12 equals zero, okay, so that's it. This is exactly what we're looking for. when you multiply y minus three by this, you're going to get this back Okay, now let's do this. So we now have that y minus three multiplied by y squared plus four y. plus 12 equals zero; this y minus three multiplied by both of them are what Equal to zero, all right. Now from here, we can say that y minus three is equal to zero. or that's y squared plus four y plus twelve equals zero, so for this side of the equation We have that our y is going to be equal to zero plus three, right? This minus three will cross over. plus three, so this means that y is equal to three. This is the first value of our y now. Recall that y is equal to what; recall where we said let x to the power of nine equal to y. so we are going to recall that x to the power of nine is equal to y, which means that we can what x to the power of nine equals three; now, how do we find the value of x? x We are going to raise each side of this equation to the power of one over nine, okay so we have x raised to the power of nine raised to the power of one over nine is equal to three raised to the power of one over nine. All right, now recall that I remember the law of indices that I explained earlier, so when you have two powers, just multiply them together, so this can give you one so you have x remaining is equal to three to the power of one over the ninth root of three, okay, now for this side of the equation remember that we are told that our y, your x, has to be real, not complex, okay? Now, for us to check if we can obtain the real root from this side of the equation; we are simply going to check whether our the determinant will give us something that is greater than or less than zero, now to check that The determinant is b squared minus 4ac, so our b is the coefficient of y, which is four. That is, b is the coefficient of y squared, a is the coefficient of y, and our constant is c, okay. so that means four squared will give us 16 minus four times our a which is one and our c is 12, now we have this; this is what 16 minus 4 times 12 will give us, 48. So you notice that this is going to give us a number that is less than zero, and that shows that this side will give us and we're told that our x is real, so this is the only real root of x, okay? Don't worry, let's Plug this into the equation to check if we're correct. So we have our x equal to three. to the power of one over nine; now let's plug it in and see if we can get the 36 on the right-hand side side of this equation, don't forget to subscribe and turn on your notification bell if this is the First time they're coming across this face, okay? So we have x to the power of what, okay? plugging in x equals 3 to the power of one over nine, right? This nine is too long. 27-calculated to the power of 27 plus three to the power of one over nine calculated to the power of 18 is equal to 36, okay. Now, when nine divides 27, you have what, three? So this side becomes three to the power of three plus when this divides this you have two, so this side becomes three to the power of two equals 36; this is 27, and this is nine, which equals 36, so this Is 36 equal to 36? The left-hand side is equal to the right-hand side. Tada, we are correct. Thank you. Thank you for watching, have a lovely day