Olympiad Mathematics | The three solutions | Can you solve this?

[00:00:01]Hi, everyone.

[00:00:04]How can you solve this problem completely?

[00:00:08]Right?

[00:00:09]So, what are you expected to do in this case?

[00:00:13]The first thing to do is to cross multiply.

[00:00:17]Yes.

[00:00:18]So, that this will multiply the zero.

[00:00:21]And we'll have x to the power of 3 minus 64 to be equal to x multiplied by zero is what?

[00:00:31]I hope you know what that is.

[00:00:33]So, we have our x to the power of 3 minus 64 and is equal to zero.

[00:00:40]So, this way almost everybody will be able to solve it.

[00:00:45]But, we know that x to the power of 3 minus um 4 to the power of 3 will be on the left.

[00:00:53]Because 4 to the power of 3 is 64.

[00:00:57]Because 64 is a perfect cube.

[00:01:00]And then imagine that we have a cube minus b cube.

[00:01:06]You know, this identity is also equal to a minus b multiplied by a squared plus ab plus b squared.

[00:01:21]Now, you have to take note of something here.

[00:01:24]And it is the fact that a is representing x and b represents 4.

[00:01:31]So, in place of a minus b, we have x minus 4.

[00:01:37]In place of a squared, we are going to have um x squared.

[00:01:44]Then, ab, that'll be x times 4 and is 4x.

[00:01:49]Plus b squared, which is 4 squared.

[00:01:52]4 squared is 16.

[00:01:54]So, we equate this to zero.

[00:01:58]Now, if you're new to this channel, consider subscribing so that you get to see more of our interesting videos.

[00:02:07]Now, the next thing to do is to apply zero product rule.

[00:02:13]Zero product rule.

[00:02:16]Now, what I'm what I'm going to do, I can't do it if this is one.

[00:02:22]If this is not equal to zero, what I'm going to do cannot be done.

[00:02:26]So, x minus four is either zero or x squared plus four x plus 16 is zero.

[00:02:38]Remember, I'm doing this because we have zero on the other side.

[00:02:43]From this part, x is equal to um zero plus four then the value of x is four. This is a solution.

[00:02:54]Now, to get the other solutions, we'll bring down this quadratic um equation there.

[00:03:01]And bringing it down, we have to bring down the coefficients and the constant, right?

[00:03:07]Remember, um a stands for the coefficient of x of x squared rather, and that is one. B stands for the coefficient of x, which is four. And [snorts] c is the constant, which is 16.

[00:03:23]Now, this a b c, for those of you that do not know, a b c are going to be found in the quadratic equation formula.

[00:03:33]Because we're going to use that very formula to solve this equation.

[00:03:37]What is the formula?

[00:03:39]Okay.

[00:03:41]So, the formula now is x equals minus b plus or minus we have b squared minus four a c all over 2 * A.

[00:03:56]So, what we'll do now is to put in our values of ABC.

[00:04:01]So, that X will now be In place of minus B, we're having minus 4.

[00:04:06]Because B is 4.

[00:04:08]Then plus or minus In place of B squared, we are having 4 squared.

[00:04:14]Minus 4 times A. A is what? 1.

[00:04:19]Times C. And our C is what? 16. So, we put 16 there.

[00:04:26]This is all over 2 multiplied by 1.

[00:04:29]Because A is still 1.

[00:04:33]Now, X is equal to minus 4 plus or minus the square root of 4 squared is 16. Then 4 * 16 is 64.

[00:04:47]So, we divide all of this by 2. Because 2 * 1 is what? 2.

[00:04:55]Yes, 2 * 1 is um 2. Now, let's continue.

[00:05:00]Now, this is one thing I know you you might want to do.

[00:05:04]You want to get the square root of 16, then minus the square root of 64. But that will be wrong.

[00:05:12]What you should do is that you take this out first before you can find the square root eventually. 16 minus 64, that will give minus 48.

[00:05:24]Then we divide all of this by 2.

[00:05:27]And now we're having negative root. So, this means that we're going to have complex solutions from here.

[00:05:33]But here is what we can do first.

[00:05:36]X is equal to minus 4 plus or minus square root of 48.

[00:05:41]Did I leave anything out?

[00:05:44]The answer is yes. I left out the negative.

[00:05:47]So, that means I should multiply everything by root of negative one.

[00:05:52]And we divide by two.

[00:05:54]But if you know very well, you will see that root 48 can be simplified.

[00:06:01]Right?

[00:06:03]Root 48 can be simplified as we have root 16 * 3.

[00:06:08]16 is a perfect square and multiplied by three, we still have 48.

[00:06:14]By the way, square root of negative one is imaginary.

[00:06:18]And then we divide both sides, we divide all through by two on the right hand side.

[00:06:24]At this point, I hope you know we can split this two.

[00:06:27]So, we have X to be minus four plus or minus square root of 16 times square root of three times I.

[00:06:37]This is all over two.

[00:06:39]Okay, I've just obeyed one of the laws of sword.

[00:06:44]Now, our X will now be minus four plus or minus square root of four square root of 16 is four.

[00:06:53]Multiply by this I, so we have four I.

[00:06:56]You know, I've not um written the root three, so we multiply by root three and we have this.

[00:07:04]Remember that all of this all of this is over over two, right?

[00:07:12]Yes, all of this is over two. Now, let's continue.

[00:07:18]From here now, our X is minus four plus or minus Um okay, let's do it this way. That is minus four over two plus or minus four I root three over two.

[00:07:34]So, if you go on, you can cancel out so that X will be two into minus four is minus two plus or minus two into 4 I that will be 2 I. Then we have root 3.

[00:07:47]So, this is a two-in-one kind of solution.

[00:07:51]Let's bring our complete solutions together.

[00:07:54]We got X to be 4 as our first solution.

[00:07:59]Then the second solution is X equals from there - 2 + 2 I root 3.

[00:08:09]Then the third solution is - 2 - 2 I then we have root 3.

[00:08:17]So, these are the three solutions to the given equation.

[00:08:23]Thank you for watching.