Olympiad Mathematics | Indian | Can you solve this?

本站添加: 2026-05-10

[00:00:00]If you're ready, let's provide the solution to this nice equation here.

[00:00:06]This is a simple one.

[00:00:10]But mind you, we are getting three solutions.

[00:00:15]We have - b cubed over 8 = 1.

[00:00:22]The first thing is to cross multiply.

[00:00:26]And that will give us - b cubed to be equal to 8 if we cross multiply.

[00:00:36]Okay, the next thing is to remove the negative from here as we multiply both sides by -1.

[00:00:44]That will make this one to be b cubed positive.

[00:00:48]And then here we have -8.

[00:00:53]Now, the next step is to bring this to the left and we have b cubed + 8 and is now equal to 0.

[00:01:04]We are not going to stop here, right?

[00:01:08]We have to express 8 in index form.

[00:01:12]As we have b cubed + 8 is 2 cubed and everything is equal to 0.

[00:01:20]And just like we always you know deal with difference of two cubes, we can equally deal with addition of two cubes.

[00:01:29]And from here, if you have um x to the power of 3 + y to the power of 3, this is addition of two cubes and is equal to x + y multiplying um x squared multiplying x squared >> [snorts] >> xy + y squared.

[00:02:01]This is our addition of two cubes.

[00:02:04]And in this case, our x is b and our y is two. So, x + y becomes b + y.

[00:02:13]Then, x squared is going to be is [clears throat] going to be our b squared.

[00:02:19]Then, we have minus xy is going to be two times b and that is two b.

[00:02:26]Then, plus y squared, our y squared is going to be two squared.

[00:02:31]And two squared is equal to four.

[00:02:34]So, we equate this to zero because of that.

[00:02:39]Now, we are multiplying two the two factors to get zero, right?

[00:02:44]So, what should we now do?

[00:02:47]We apply our zero product rule.

[00:02:55]We apply our zero product rule because we are multiplying two terms to get um zero.

[00:03:02]So, what do we do?

[00:03:04]We will say that it's either b + y is zero or b squared minus two b plus four is equal to zero.

[00:03:20]And from the left-hand side, we can get our Okay.

[00:03:29]We can get our Okay, my bad. My bad. By the way, this is not b + y.

[00:03:35]It is rather b + 2.

[00:03:39]Because we say that x is b and our y is two.

[00:03:44]So, it is b plus two.

[00:03:48]Okay. So, here we have the same B plus two.

[00:03:54]And from here, it means that B is going to be zero minus two.

[00:03:59]And B is equal to minus two.

[00:04:03]We have our solution already from there.

[00:04:07]>> [snorts] >> But we need to get two more solutions.

[00:04:11]And that will come out of this quadratic equation.

[00:04:14]So, let's transfer this equation very quickly.

[00:04:21]Okay, so we're going to use our quadratic formula to solve this.

[00:04:26]Um the formula I want to use has ABC.

[00:04:30]You know what? I will change this very quickly. So, let this be x squared minus two x plus four equals zero. But know that your B is still the x that we have here.

[00:04:44]And what is the formula? The formula is x equals minus B plus or minus the square root of B squared minus four AC all over two times A.

[00:05:00]So, we will now put in the values of ABC.

[00:05:04]A is the coefficient of x squared.

[00:05:08]B is the coefficient of x and C is a constant.

[00:05:12]So, here now we're going to have minus minus two.

[00:05:17]Because B itself is minus two.

[00:05:20]Plus minus B squared as minus two squared.

[00:05:25]Then minus four times one times four.

[00:05:29]Because A is one and C is four.

[00:05:34]This is over two times one.

[00:05:38]And to go on with this, we have our x to be negative negative is positive. So, we write two.

[00:05:47]plus minus minus two squared is minus two times minus two that is four then minus four times four is 16.

[00:05:57]So we write our 16.

[00:06:00]And we are dividing this by two.

[00:06:03]So to go ahead now we have X to be two plus minus four minus 16 is minus 12.

[00:06:12]And we get our two on that.

[00:06:17]So we're going to get two plus minus square root of 12.

[00:06:22]Oh, I didn't pick the negative, right?

[00:06:25]So that means I should multiply this by negative one.

[00:06:28]And divide this by two.

[00:06:32]But we know that X from here is two plus minus square root of 12 can be written as square root of four times three then times square root of negative one which is not real. It is imaginary and we write I.

[00:06:51]Divide this by two.

[00:06:54]So you can always split this to get two plus or minus square root of four times square root of three times I.

[00:07:06]And it's over two.

[00:07:09]So from here now we can go ahead to get our X to be equal to um two plus or minus square root of four is two.

[00:07:20]Two times I is two I then we have root three. This is divided by two.

[00:07:26]So to go on with this we're going to have X to be equal to two can go into two we have one plus or minus two into two two into two I is going to be I.

[00:07:40]Then we have root three.

[00:07:42]Mind you, this is a two-in-one solution because of the plus or minus. So, let's bring the three solutions together since we had one solution before.

[00:07:57]Okay, so let's bring down the three solutions to this very equation.

[00:08:02]The first one, we have B to be equal to minus one.

[00:08:09]Okay?

[00:08:11]And then we we are also having um Okay. From the last part, we are having that um X, you know, we solved for X in the last part, but that X represents B. So, our B2 now, the second solution is one plus I root three.

[00:08:38]Right? Then we get our third solution, B3, which is equal to one minus I then we have root three. So, these are the three solutions to this very equation.

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