Olympiad Mathematics | Germany | Can you solve this?

Added: 2026-05-05

[00:00:01]If you are told to provide the three solutions to this equation, what would you do?

[00:00:09]The equation is x to the power of three divided by eight equals one.

[00:00:17]This is the equation and you are to get three solutions.

[00:00:20]It looks impossible, right?

[00:00:24]But it is very, very possible. What do you do? Cross multiply.

[00:00:30]So that you have x to the power of three on this side.

[00:00:34]Then you have eight times one, which is eight.

[00:00:39]Now, you can see that it's very possible to get three solutions.

[00:00:44]But do not find the cube root of both sides. If you do that, you will end up having just one solution.

[00:00:50]So what do you do rather?

[00:00:53]You will rather write eight in index form, which is two to the power of three.

[00:00:59]So by now, you know what we'll do, right?

[00:01:02]Okay, so we have x to the power of three minus two to the power of three.

[00:01:08]This is equal to what?

[00:01:10]Zero, because this has been shifted to the left and there's nothing on the right hand side.

[00:01:17]So to take a step further, we apply our difference of two cubes.

[00:01:23]We know that imagine that you have m to the power of three minus n to the power of three.

[00:01:31]This is equal to m minus n into m squared plus mn plus n squared.

[00:01:48]Okay, so this is an identity. Let me show it. This is an identity in mathematics.

[00:01:55]Our m is x and our n is two. So let's put them into this.

[00:02:01]We have x minus two into we have x squared plus two x. That is mx.

[00:02:13]x times two, two x.

[00:02:15]Then plus n squared, which is two squared.

[00:02:18]Two squared is four. So this is equal to zero.

[00:02:23]Now, from here we know what to do. We're going to apply what we call zero products.

[00:02:34]Zero product rule.

[00:02:37]When do we apply this rule? We apply this rule when we are multiplying two terms to get zero.

[00:02:44]So we believe that one of them must be equal to zero.

[00:02:48]If not, the right hand side will not be equal to zero.

[00:02:53]So to continue with that we will say that our x minus two is either zero or x squared plus two x plus four is equal to zero.

[00:03:12]Okay, so I'm going to focus on the left for now and I will get x to be zero plus two if I collect like terms.

[00:03:23]Okay, and now the value of x is equal to two.

[00:03:28]This is one of the solutions.

[00:03:32]Now, we are going to get two more solutions from this side.

[00:03:36]So how do we go about that? Let's bring it and then solve it.

[00:03:43]Okay, so this is it. Let's solve it now to get the three solutions.

[00:03:48]We um to get We are getting two solutions from here, by the way.

[00:03:52]Now, our a is one, our b is two, and our c is four.

[00:04:01]a b c, they are the you know coefficients of x squared coefficient of x and the constant respectively.

[00:04:13]So what then is the equation? I mean, the formula we are going to use.

[00:04:17]The formula is x equals minus b plus or minus the square root of b squared minus four a c.

[00:04:32]Everything is over two multiplied by a.

[00:04:36]So what do you do at this point? Put in the values of a b c.

[00:04:42]So that we can have x to be minus two, because b is two.

[00:04:49]Then we have plus or minus b squared. That's going to be two squared minus four times one. a is one, then times four because c is four.

[00:05:04]This is all over two multiplied by one, which will give us two eventually.

[00:05:11]So our x now is minus two plus or minus four minus By the way, do not say four minus four so that you get four here.

[00:05:24]Two squared is four, then four times four 16. So you have to use both minus in there as well.

[00:05:35]This is over two. So that's our x from here will be equal to minus two plus or minus the square root of minus 12. Four minus 16 is minus 12.

[00:05:48]And this is over two like we all know.

[00:05:52]Now, we are going to simplify this root negative 12. The first thing is to remove the negative from there.

[00:05:59]So we have this. We have 12. Now the negative is gone.

[00:06:04]But now that the negative is gone, it has to you know, be multiplied so that the answer will not change. The solution will remain intact.

[00:06:15]This is all over two.

[00:06:19]Now, you look at that 12 again. There's a perfect square that is a factor of 12.

[00:06:25]So bring out the perfect square in there.

[00:06:28]Plus or minus the perfect square there is four.

[00:06:31]Four times three will give 12.

[00:06:33]Then square root of negative one is i, imaginary.

[00:06:37]And everything is over two just like we can see.

[00:06:42]And here we are multiplying these two.

[00:06:44]There's one of these law There's one law that says you can split what you have here.

[00:06:51]Let me do it here.

[00:06:52]x is equal to minus two plus or minus we have square root of four multiplied by the square root of three multiplied by i.

[00:07:04]Okay, can see that that is correct.

[00:07:06]So everything will now be divided by two.

[00:07:09]So we go.

[00:07:11]Our x will be minus two plus or minus square root of four is two. Two times i is two i, then times root three.

[00:07:23]Remember that this is still over two.

[00:07:28]Okay, so if if we go on we will now have our x to be equal Okay, so our x is minus one. Two into minus two plus or minus two into that is going to be i. We have our root three.

[00:07:45]Now, this two is gone.

[00:07:48]Let's bring the three solutions together now. Remember we got x to be two as one of the solutions.

[00:07:56]Then we are having x two.

[00:07:59]This is x two. This is x one.

[00:08:02]x two is from here, which is minus one plus i root three.

[00:08:09]This is our x two.

[00:08:11]Now, our x three, the third solution, is minus one minus i root three.

[00:08:19]So these are the three solutions. Thank you for watching.

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