Olympiad Mathematics | Germany | Can You Solve This?

Añadido: 2026-05-09

[00:00:01]Hi, everyone.

[00:00:04]If you're ready, let's solve this equation in total detail.

[00:00:12]X to the power 3 equals the square root of 64.

[00:00:18]And 64 is a perfect square.

[00:00:21]So, we have X to the power of 3 to be equal to the square root of 64, and that is 8.

[00:00:31]We want to get the complete solution to this equation. So, we're getting three solutions.

[00:00:39]And our X to the power of 3 is equal to 8, and the 8 there is 2 to the power of 3.

[00:00:48]So, now we'll bring 2 to the 2 to the power of 3 to the left-hand side, so that X to the power of 3 minus 2 to the power of 3 is equal to 0.

[00:01:02]And this is um difference of two cubes.

[00:01:06]I will know that if we have a cube minus b cube, this is expressed as a plus b a minus b, right?

[00:01:18]Then multiply by a squared plus ab plus b squared.

[00:01:26]Difference of two cubes.

[00:01:28]And according to this standard, our a is x and our b is 2. So, a minus b now will be x minus 2 into a squared, which is x squared, plus ab, that is x times 2, then plus 2 squared.

[00:01:48]Because b is 2, and all of this is equal to 0.

[00:01:54]From here, we work on the second bracket.

[00:01:58]X minus two is over here.

[00:02:01]Then X squared plus X times two is two X.

[00:02:05]Then two squared is four.

[00:02:08]This is equal to zero.

[00:02:11]So, we are going to apply our zero product rule.

[00:02:20]We apply this rule because we are multiplying two terms together zero. So, either of them will be equal to zero. So, we are saying that X minus two is equal to zero or X squared plus two X plus four is equal to zero.

[00:02:41]Now, working on the left-hand side, what we have here our X is going to be zero plus two meaning that X is equal to two.

[00:02:52]This is our first solution.

[00:02:55]Now, we are going to need two more solutions from here.

[00:02:59]And that will come out of this.

[00:03:01]By the way, this is a quadratic equation because of the power of two the unknown variable X and the equal sign.

[00:03:11]Okay, these are the three conditions that make an equation to be a quadratic equation.

[00:03:17]The unknown variable, the power of two, and the equal sign, right? So, let's pick a pick the equation and then solve it.

[00:03:28]Okay, so we have our quadratic equation now.

[00:03:34]And we want to solve it using the quadratic formula.

[00:03:38]The formula has A B C, right?

[00:03:42]The A is a coefficient of X squared, which is one.

[00:03:46]>> [snorts] >> Then B is a coefficient of X, which is two.

[00:03:51]And then the C is the constant which is four.

[00:03:57]So, what is the formula then? The formula is X equals minus B plus or minus B squared minus four AC all over two times A.

[00:04:12]So, what we're going to do now is to put the value of ABC into the formula.

[00:04:18]So, that our X will be minus two because B is two, right? Then plus or minus B squared which is two squared.

[00:04:29]Then we have minus four times A times C.

[00:04:34]A is one, C is four.

[00:04:37]And it is all over two times one.

[00:04:42]So, from here now we get our X to be minus two plus or minus we have two squared is four. Then four times one times four is 16.

[00:04:55]And we're dividing by two.

[00:05:00]Now, the value of X is equal to we have minus two plus or minus then the square root of four minus 16 is minus 12.

[00:05:12]And this is all over two.

[00:05:16]From here, our X is going to be minus two plus or minus square root of 12 multiplied by square root of negative one.

[00:05:27]Okay, so I had to do this so that we can easily get the square root of 12.

[00:05:33]And um this is all over two. In case you do not know if you multiply root 12 by root negative one, you're going to get root negative 12.

[00:05:45]Now, let's work on the root 12 there.

[00:05:49]Root 12 of here can be, you know, split to get root 4 * 3 then multiply by I cuz square root of -1 is I. 4 * 3 is 12.

[00:06:03]And we still have this over 2.

[00:06:06]Now, since we can split this, we can find the square root of 4 separately and then find the square root of 3.

[00:06:14]So, our X is -2 plus or minus square root of 4 will give us 2.

[00:06:20]Multiply this 2 by this I, so we're having 2i.

[00:06:24]Then root 3 will come down here and it's all over 2.

[00:06:33]Okay. So, from here now, what should we do?

[00:06:36]We can reduce this equation this value.

[00:06:40]By the way, 2 is a common factor. So, we can factor out 2 from the numerator.

[00:06:48]So, if 2 comes out, we have -1 from here then plus or minus I because 2 is already out, then we have root 3.

[00:06:59]This is all over 2.

[00:07:01]So, this one here can go with this.

[00:07:05]And that means that our value of X is um -1 plus or minus we have I root 3.

[00:07:17]And in case you do not know, this is a two-in-one solution because of the plus or minus.

[00:07:23]So, this means that our X is equal to -1 plus I root 3 or -1 - I root 3.

[00:07:36]Now, we're going to bring the three solutions together.

[00:07:40]Okay, I'm going to write the three solutions here.

[00:07:43]X1 that we got before now is equal to two.

[00:07:47]Then X2 is from here, this particular one.

[00:07:52]And it's minus one plus I root three.

[00:07:59]Right? That's our second solution. Then the third solution X3 is equal to minus one minus I root three. So, these three are the solutions to the equation.

[00:08:16]Thank you for watching.

[00:08:18]Remember this is minus one.