Math Olympiad| I enjoyed solving this math Olympiad algebra question.

Can you get the three values of x? Something we can do very quickly. Let's get it. Okay, now. first let's move this part of the equation to the other side of the equation, so if we do that so we'll have x to the power of three, I don't even know why I'm writing my x like this Okay, x cubed plus x minus 10 equals zero. Yes, because if plus 10 crosses over Now look at what we have here—we have a cubic equation, right? But in this cubic equation the second degree is not present, nor is x to the power of three, but nevertheless we We can still solve for x; first of all, we need to try, and we need to use trial and error to solve this. We need a number that, when substituted for x, still gives zero in this equation. and if you try one minus one two, and as you notice that two is the right option, so if we say let x equal 2, you will have that 2 to the power of 3 plus 2 minus 10 is equal to zero now two to the power of three is eight plus two minus ten is equal to zero so you bead Eight plus two is ten, minus ten equals zero, and this is true, so that means x equals two. is what a factor is, this thing is a root, sorry, now we need to use synthetic division to get the other quadratic equation, okay, so to do that we need this; it is synthetic division, we are going to use these two to divide the coefficients of what we have here, which is one x cubed, remember that we're supposed to have x squared, as this is not there, that means the coefficient is zero, then we won't have one x and our constant is minus 10, so this is how it goes—we are going to bring down one Now when two times one, two times one, we get two. We add zero plus two, we get two. Now two. times two is four, we add one plus four and we get five. Two times five is ten, we add and you'll have zero once you get zero after dividing like this simply means that the number being divided That's what a root is; with this, we have obtained the coefficients of our quadratic equation. Alright, if you want to know in depth how to solve quadratic equations and how to solve cubic polynomials. I'm going to leave the link as it is, as an end screen at the end of this video. I have a detailed tutorial that will give you three different methods you can use to solve Any type of cubic equation, okay, so let's continue with the cubic equation or cubic expression. uh, cubic polynomial, sorry. Now look at this; so that simply means that if x equals two is a root So this implies that x plus two crosses here becomes minus two, and that is equal to zero, right? so these are what we have: that x minus two is a factor, and when it's multiplied by this, so we have one x cubed, one x squared, right? So this is x squared plus two x plus two x plus five. plus five equals zero, so we multiply this and this, and we will get this equation back. Perfect, now let's continue. So when you have two numbers, or two expressions, or equations, or whatever, when two things, when you multiply two things and you get zero, that simply means that either of the two numbers is zero, this is zero, so we can say that x minus two is equal to zero, and that x squared plus two x plus five equals zero. Remember that we are going to get three different roots, since this is degree three, right? So three values of x. So the first value of x is that x equals when minus two crosses the equation it becomes plus two, so this is x. One, all right, now let's get to this side of the equation. When you look at this, notice that you cannot be solved by the factorization method, but we can use the quadratic equation formula to find the solution. the value of our x; now, the quadratic equation formula states that this x is equal to minus b plus or minus the square root of b squared minus 4ac, divided by 2a. All right, now in this case, in this case Our a is the coefficient of x squared, which means that a is equal to one. b is the coefficient of x. That means b is equal to two, and our c is a constant, which is plus five, so let's substitute. into this particular formula, so we have that x is equal to minus b, that is minus two plus or minus the square root of b squared, which is two squared, minus four times a, a being one and c is what five everything divided by 2a, two times a is what one perfect, so we now have that x is minus two plus or minus the square root of two squared, which is four, minus four times five, which is 20. everything divided by two, okay? Now, if this is the first time you are coming across This lovely channel, please click the subscription button and the notification bell so that you don't miss anything. I will be notified anytime I upload a new video. Thank you so much. I upload two videos daily. Sometimes three, so I will suggest, I'll plead: always switch to my page to see what's new. Alright, okay, so let's continue. Now this is what we have, so that means that x is equal to negative two plus or minus square root of negative 16, right? 4 minus 20 is negative 16 divided by 2, okay, so we have that x is equal to minus 2 plus or minus, so this same thing as square root of 16 multiplied by square root of what minus 1 divided by 2, so our x is minus 2 plus or minus the square root of 16 is 4, and the square root of minus 1 is i, divided by 2, so we can write that x is equal to minus 2 divided by 2 plus or minus 4i divided by 2, all right now when this divides this, what will you have? We have that x is minus 1, right? Plus or minus. When this divides this. So let's find the second x₂; this means that x₂ is minus 1 plus 2i and x₃ is minus 1 minus 2i. So these are the three values of our x. Thank you so much for watching, and have a lovely day.