Math Olympiad Algebraic math question| Many got this wrong| solve for x.

Now let's solve for the value of x to eliminate this 27, because I don't want a fraction here. I'm going to multiply and multiply through by 27, so when I do that, I will have 27 multiplied by x. Before 3 plus 27 multiplied by x before 2 equals 4, okay. All right, now 27 is the same thing. I can write this as 3 raised to the power of 3 multiplied by x to the power of 3 plus 3 raised to the power of 3 multiplied by x before 2 is equal to 4, all right? Now recall that when you a to the power of m times b to the power of m is the same as a times b raised to the power of m. All right, so that means we can rewrite this as 3x to the power of 3 plus 3 to the power of 3 but we can relate or we can rewrite 3 to the power of 3 as what 3 multiplied by 3 to the power of 2, right? Yes, because a to the power of m plus n is equal to a to the power of m multiplied by a to the power of n, so this is the same thing as 3 to the power of 1 plus 2, which will still give us 3 to the power of 3, back all right, so it's equal, okay, multiplied by x squared is equal to 4, now. we can write this as 3x³ plus 3 multiplied by 3x² is Equal to 4 now we can say let 3x be equal to something; you can say let it be equal to y. and when we do that, we can now write this equation as yes, I know this is where some of you will be Yes, we have formed our quadratic equation, but it doesn't stop here, okay? So even though we are I'm going to form a polynomial now; I'll say, advice you watch to the end to see how we got our x from. what we just assigned y; now we have this: we say let 3x be equal to y, so this is y to the power of 3 plus 3y squared, then this 4, let's bring it in minus 4 is equal to 0, okay. All right, now if this If this is your first time encountering this channel, please don't forget to give us a thumbs up. then on your notification bell so that you'll be notified anytime we upload a new video, okay so we have y to the power of 3 plus now 3 is the same thing as 2 plus 1, right? 2 plus 1 multiplied y squared minus 4 equals 0. Now let's expand the bracket; we have y to the power of 3. plus 2y squared plus y squared, okay, minus 4 is equal to 0. All right, now I think we can now Solve by factorization, okay, because when you look at this and this, we can factor out y, okay, so When we say y bracket y to the power of 3 divided by y we get y squared plus 2 y squared divided by y we are going to have 2, okay, sorry, we can factorize y squared out, not just y; y squared is so let's factor out y squared, so what we do is we have y squared in brackets times y to the power 3 divided by y squared gives us y plus 2; y squared divided by y squared gives us 2. All right, plus now we have y squared; this is minus 4, right? Is equal to 0. Now we have y squared. bracket y plus 2 plus we can write this as y squared minus 4 is the same as 2 squared is equal to 0 now I want us to recall look at what we are seeing here this is difference of Yes, so we can write this as: when you have a squared minus b squared, this is equal to a minus b multiplied by a plus b, okay, since this is equal to this multiplied by this, we can now rewrite this as so we have y squared times the quantity y plus 2, and we can now write this as y minus 2 multiplied by y plus 2 is also equal to 0. Now when you look at these two terms You can notice that we have y plus 2 here and we also have y plus 2 here, so let's factorize that. so we have y plus 2 open bracket when all of this is divided by this we have y squared The remainder plus when this is divided by this gives y minus 2, and the remainder is equal to 0, okay all. Right now we can now write two things: that y plus 2 is equal to 0, and that y squared plus y y squared plus y minus 2 is also equal to 0. Now, from the first part of the equation record that we are not looking for the value of y, we are looking for the value of x, but we are I'm going to make use of this y to get the value of x, okay? So we have from here that y is equal to minus Then from this side, from this side, we have to use the factorization method. Yes, we can, because What are the two factors of minus 2 that, when added together, give plus 1? be 2 and minus 1, right? Yes, so we can now write this as y squared plus 2y minus y minus 2 is equal to 0, all right, so we can now solve this side as well when we factorize this and this Do they have in common? They have y in common. Now, y squared divided by y gives us y plus 2y. Divided quis by y we get 2, okay, then minus 1 in parentheses, which can also be written as y plus 2. is equal to 0, yes, because minus y times minus 1 times y still gives us minus y and minus 1 times plus 2 still gives us minus 2, okay? So from here, we can now say that y plus 2, this is what we have here and here is equal to 0, and that's y minus 1 is also equal to 0, so we can see that our second value of y, this is the first value of y, this is the second value of y is equal to minus 2 again means we have minus, we have minus 2 twice, then the third value of y Remember that we have not yet found x, so we are still obtaining y; therefore, do not get confused. Now minus 1 will carry over to become 1, so these are the three values of our y—the first value of y That's what y is; these are the values of y we have: minus 2, minus 2, and 1, so these are the values. of our y, okay, now we are going to take each of them one by one and solve for x. Now recall that our x is what we said we said I'm looking for where we assigned this to this, so we said let 3x be equal to y, right? I'm sorry, I didn't even write it down. We said let 3x be equal to y, so it's simply means that I'm going to clean here, so I will write: recall that 3x is equal to y, okay, so that I won't forget now we have three values of our y: they are minus 2, minus 2, and what, 1? So if 3x is equal to y, so for y equals minus 2, all right. So we can say that 3x is equal to minus 2. Now to get The value of x we divide by 3, divide by 3, and we have that x is equal to minus 2 over 3 now this It's going to be twice, that's fine, since we have y equals minus 2, twice, right? Recall minus 2. minus 2; now the other value of x is what when y equals 1? Right, when y equals 1, we can write that So divided by x, sorry, divided by 3, divided by 3, and we have that our x is equal to 1 over 3 now. These are the three values of our x. Now let's check to see if we are right. Now the first value states that our x is what 1 over 3; let's use this to check, so we have 1 over 3 to the power of what plus I'm substituting here now plus 1 over 3 to the power of 2 is equal to 4 over 27 now 1 to The power of 3 is 1; 3 to the power of 3 is 27; plus 1 to the power of 2 is 2; is 1; and 3 to the power of 2 is 9 is equal to 4 over 27; now we have 127 plus 1 over 27; plus, now this is telling us we have to multiply the denominator by 3 to get 27, so it's going to be 9 times 3 is 27, so we also multiply the numerator by 3: 1 times 3, which still gives us what? 1, sorry. is equal to 4 over 27; now that we have the same denominator, we just add the numerator, sorry What did I do wrong here? What did I do wrong here? Sorry, 1 times 3 is 3. What am I saying? I knew something was wrong somewhere, yeah, record that to get this: 9 times 3 gives us 27, so 1 times 3 gives us 3, not 1. So 1 plus 3 gives us 4, so we have 4 over 27. is equal to 4 over 27, I'm also sure that when you substitute your x to be equal to minus 2 over 3, you're also going to get 4 over 27 on the right-hand side of your equation Thank you so much for watching, have a lovely day.