A Nice math Olympiad algebra question| solve for all values of x

of x. So the first thing we need to do is let's bring them together. So when this crosses becomes minus x minus one to the power of four okay is equal to zero. Okay let's split this as x to the power of two raised to the power of two minus x minus one to the power of two as well is equal to zero. Yes because a to the power of m raised to the power of n is equal to a to the power of m times n. Alright okay so let's continue. Now we have a square this minus this squared minus this squared therefore we have a squared minus b squared and according to the identity same thing as a minus b multiplied by a plus b. So in this case our a is x squared and our b is x minus one squared. So it's going to be x squared minus x minus one squared okay multiplied by x squared plus x minus one squared is equal to zero. So at this point we have two this thing right. So at this point we have two values or two parts of our equation. So the first one is what x squared minus x minus one squared is equal to zero. That is the whole of this is equal to zero and the second one is x squared plus x minus one squared is also equal to zero. Now we solve from the left hand side we have x squared minus now when you have a minus b squared according to our algebra identity a minus b all squared is equal to a squared minus 2 a b plus b squared. That is our algebra identity therefore we are going to express this in this form okay. In this case our a is x and our b sorry in this case our a is x and our b is one okay. So we are going to have this going to be x squared so we have x squared minus 2 a b that is two multiplied by x multiplied by one plus b squared that is one squared is equal to zero. So we're going to focus on this first case first this case one so let's call this side case one. So that is exactly what we are doing now. So if we expand this more we have x squared minus so we have x squared we have minus 2 x right minus 2 x and we have plus what plus one because one squared is one is equal to zero. Now I'm going to rewrite this second case somewhere so that I can easily clean the board and we get back to it at the due time. So x squared plus x minus one squared is equal to zero. All right now I can clean this because I need more space. So if this is the first time you are coming across this channel please subscribe all right and turn on your notification bell please so that you don't miss our next video. Now we have x squared minus times x squared will give us minus x squared minus times minus 2 x will give us plus 2 x minus times plus one will give us minus one is equal to zero x squared minus x squared is zero so we are left with 2 x minus one is equal to zero. So 2 x is equal to let's transpose this one to the other side if we do that we have one right becomes plus one so let's divide by two divide by two and we have that x is equal to half so this becomes the first value of our x. Now recall the second case according to the second case second case says that x squared plus x minus one squared is equal to zero. Recall I wrote it somewhere so we have x squared plus now if we open this bracket it's going to be x squared right minus 2 a b that is two times x times one which is 2 x plus b squared which is one squared which is still give us one is equal to zero so we have x squared plus x squared will give us 2 x squared minus 2 x plus one is equal to zero. Now at those points we are going to make use of our quadratic equation formula to solve for the value of our x have you given us a thumbs up yet if you have not please give us a thumbs up if you're actually enjoying our video all right so that you encourage the algorithm to push our video to more people that need to see it okay so we are going to say x is equal to minus b plus or minus square root of yeah that this is the quadratic equation formula now our a is coefficients of x squared which is two our b is coefficients of x which is minus two and our c is our constant which is one so we are going to substitute this into this particular equation and we are going to have that x is minus minus two right plus or minus square root of minus two squared okay minus four a c our a is two our c is one all right everything divided by two a that is two multiplied by two okay so we have that x is equal to minus minus becomes plus plus or minus square root of minus two squared is four minus four times two is eight all right everything divided by two times two is four all right so we need to clean this side of the board we found the fourth value of x as one over four right okay i'll be one over two sorry now we have that x is equal to two plus or minus square root of minus four right yes because four minus eight we give us minus four divided by four so we have x is equal to two plus or minus excuse me now this root uh minus four is same thing as root four multiplied by root minus one all right divided by four so we have x is equal to two plus or minus square root of four is two and square root of minus one is i divided by four now when you look at the numerator we can factorize two out so we bring out two two divided by two we give us one plus or minus two i divided by two we give us i divided by four now we can write sorry no no one entered my mouth okay so we have that x is equal to when two divide itself we have one okay when it divides four we have two therefore we have one plus or minus i divided by two left right one plus or minus i divided by two therefore the next value of our x is one plus i divided by two and the next value is one minus i divided by two so these are the values of our x thank you so much for watching