Math Olympiad|Solve for x|Algebra

Hi, let's solve for x first. Let's factorize this. Okay, when we do that, we are going to have x squared bracket when this is divided by this we have one plus when this is divided by this we have one divided by x plus one squared, okay? Now this is equal to three. Now before we proceed because we are going to expand something inside, okay, let's just continue now we have x to the power of two multiplied by now let's add what we have inside the brackets the lcm is um x plus one to the power of two, right? So we have x plus one squared. Now, when one divides x plus one squared we have x plus one squared times this one, we still have the same thing, x plus one squared and now when this is divided by this, we get what? When this is divided by this, we get one. One times one, we give us one, close brackets, is equal to three. Now I want us to recollect. Please recall that a plus b raised to the power of two is equal to a squared plus two a b. plus b squared, okay? So in our case, our a is x and our b is one, so let's expand. So we have x squared. (x + 1)² is the same thing as x² + 2 multiplied by x multiplied by one right plus our b squared is one squared then plus this one here now everything divided by x plus one squared. All right, this is equal to three. All right, now we have x to the power of two open brackets so we have x squared plus two x plus what one squared is one so one plus one we give us two divided by x plus one squared and this is equal to three oh you are not seeing that, so all I did was add this to this, so x squared is x squared plus two times x is two x plus one squared is one plus one is two, so that is why I got what I got, okay? So we have x squared divided by x plus one squared Okay, so then I want to combine this: two x plus two divided by x plus one squared. Okay, this. is equal to three, all right? Now let's continue. Now when you look at this, we can actually see Something here, right? Okay, so we have x to the power of two; I'm talking about this place. We open our brackets; this is x squared divided by x plus one squared, okay, plus now we can factorize. two from here, so we have two brackets x plus one divided by x plus one squared, okay, this is equal to three, okay, now we have x squared, into we have x squared divided by x plus one squared plus now Here we have one x plus one, and we have double of it, so when we cancel one, it remains as the denominator. So we have two divided by x plus one remaining, okay, and this is equal to three. Now let's open this bracket, so x squared times x squared will give us x to the power of four divided by um x x squared plus one squared, right, plus two times x squared, will give us two x squared divided by x plus one yeah, divided by x plus one alone, and this is equal to three. All right, now, um, we can express x to the two divided by x plus one squared as well, okay, plus then here we have two x squared over x plus one is equal to three. Now when you have a to the power of m a divided by b to the power of m is equal to a over b raised to the power of m. Alright, so we can solve this one as x squared divided by x plus one. everything raised to the power of two, okay? All right, then, plus this is two brackets. x squared divided by x plus one, all right, and this is equal to three. Now we can see x squared divided by x plus one here, and x squared divided by x plus one here; therefore, we can say that Let x squared divided by x plus one be equal to y, and when we say that, that means what we have. here is y squared plus what we have here is two y equals three, so what we just found is a quadratic equation, but do not be in a hurry to leave, because recall that we are not looking for The value of y is what we're looking for, so please stay put and see how we obtained x from y. All right, thank you so much. Now, when you look at this, let's bring this in so we have y squared. plus two y minus three equals zero; now, if we compare this with the standard quadratic equation Here, a is the coefficient of y squared, b is two times the coefficient of y, and c is minus three. Okay, so we are going to solve this using the quadratic equation formula. This states that this What we are looking for is equal to minus b plus or minus the square root of b squared minus 4ac. Everything divided by 2a, okay. All right, permit me to continue from here. Let me draw my line first. Okay, so let's continue now. This is where we are: that our y is equal to minus b plus or minus square root of b squared minus 4ac, all divided by 2a, and our a is equal to one, b is two and our c is minus three, okay. Now let's continue. So our y then will be equal to minus b, that is negative 2 plus or minus the square root of b squared, which is 2 squared minus 4ac, 4 multiplied by 1 multiplied by minus 3, right? Everything divided by 2, multiplied by 1. All right, so we have that y. is equal to minus 2 plus or minus square root of 4 then 4 times minus 3, which gives us minus 12 minus minus becomes plus; 12 divided by 2. All right, now we continue. So when you look at what Here you can see, sorry to interrupt, yes, so y is equal to minus 2 plus or minus square root of 16 divided by 2 okay all right so we have y is equal to minus 2 plus or minus 4 divided by 2; now we have two values of our y. The first value is minus 2 plus 4 divided by 2. and the second value is minus 2 minus 4 divided by 2, so the first value of y is minus 2 plus 4 is something that's 4 minus 2, which is 2, and 2 divided by 2 is 1, and the second value of y is minus 2 minus 2, which is minus 4, I'm sorry, which is minus 6, and minus 6 divided by 2 we give US minus 3, all right. Now we have two values of y, but I want you to recall, but recall where we said. Let y be equal to x squared divided by x plus 1. This place, yeah, okay, yeah. All right, so now we are let y equal x squared divided by x plus 1; now the first value of y is 1, right? So we have 1 is 1. We cross-multiply and obtain x squared equals x plus 1. Therefore, we have x squared minus x. -1 is equal to 0, so if we use the quadratic equation formula to solve this, we have that x is minus b plus or minus the square root of b squared minus 4ac, all divided by 2a. Now in this Case our a is equal to 1, right? Our b is minus 1, and our c is also minus 1. Now let's substitute. So x is minus minus 1, which is 1, plus or minus the square root of b squared, that is minus 1 squared. which is 1 minus 4ac, that is 4 times 1 times minus 1, which will give us minus 4 minus minus is plus, so we have plus 4 here, everything divided by 2a, which is 2 times 1, which is 2. So x is 1 plus or minus the square root of 5. I'm trying to be faster; I don't want this video. So it's too long, okay, all right, so this gives us the first and the second values of our x. Now let's go back again, okay? So if we go back again, we will recall that the second value of y What minus 3? So when y equals minus 3, let me see if I can make use of this remaining space. Please allow me to proceed now. When y is equal to minus 3, this is what we have. trying to save this row, okay, all right, now when y is equal to minus 3, okay, so we have that x is equal to okay that our y is equal to x squared divided by x plus 1 this is what our y is so if y equals minus 3, so definitely minus 3 equals x squared divided by x plus 1, all right. Sorry about that, I'm a bit distracted. Okay, let's go now. We are going to cross-multiply, so we have x squared is equal to minus 3x minus 3, so if we move this to the other side, we have x squared plus 3x plus 3 equals 0. Now we are going to use our quadratic equation formula, which says that x is equal to minus b plus or minus the square root of b squared minus 4ac, all divided by By 2a, all Https://www. Ideal. com/ 1234567890 Also 3, now let's solve for x; then it should be equal to minus 3 plus or minus the square root of 3 squared. is 9 minus 4ac, 4 times 1 times 3 is 12, divided by 2 times 1, which is 2, all right, so x is minus 3 plus or minus the square root of minus 3 divided by 2, so x is minus 3 plus or minus the square root of 3i divided by 2, so these are the third and fourth values of our x. Thanks so much for watching. Please click the subscription button and turn on your notification bell if this is your first time. And please give us a thumbs up if you actually enjoyed this video, bye