Olympiad Mathematics | Indian | Can you solve this?

Hinzugefügt: 2026-05-05

[00:00:01]Hi.

[00:00:03]Let's provide the complete solution to this problem here.

[00:00:10]x cubed over 2 equals 4.

[00:00:16]Okay. Now, when I say complete solution, I mean the three solutions that will satisfy this equation.

[00:00:24]Why three? Because of the power of three.

[00:00:28]So, what we'll do is to cross multiply.

[00:00:31]And um if we do that, we'll have x to the power of three times one.

[00:00:37]That will still give us x to the power of three.

[00:00:40]Then two times four, that will give us eight, right?

[00:00:45]Now, don't take the cube root of both sides, rather you let them have the same powers.

[00:00:53]So, we write x to the power of three to be equal to two to the power of three, cuz that's the same as eight.

[00:01:01]Now, bring the two to the power of three to the left.

[00:01:06]So, we have x to the power of three minus two to the power of three now equal to zero.

[00:01:16]From this point, we shall apply what we call um difference of two cubes.

[00:01:22]It's an identity in mathematics.

[00:01:25]And it says that a cubed minus b cubed is equal to a minus b multiplied by a squared plus a b plus b squared.

[00:01:43]Okay, so this is an identity.

[00:01:47]And we're going to use this identity to solve this problem now.

[00:01:51]All right, a is x and our b is two.

[00:01:55]So, we have x minus two.

[00:01:59]Then, a squared is x squared.

[00:02:03]a b is 2 x.

[00:02:06]That is two times x.

[00:02:09]Then plus b squared. And that um that is two squared, right?

[00:02:14]So, we put two squared as b squared.

[00:02:17]Now, everything here is equal to zero.

[00:02:23]From here, we have our x minus two into here, this is x squared plus 2 x plus four to be equal to zero.

[00:02:38]So, we can now apply what we call zero products rule.

[00:02:49]We apply this rule whenever we multiply by multiply two terms together.

[00:02:56]So, at this point now, we are saying that x minus two is equal to zero or x squared plus 2 x plus four is equal to zero.

[00:03:09]But, we're going to work on the linear part, the linear equation first before we talk about the quadratic.

[00:03:18]From here now, our x is zero plus two because the additive inverse of minus two is plus two.

[00:03:26]>> [snorts] >> Now, our x is already two.

[00:03:31]So, this is one of the solutions.

[00:03:34]And to get the other solutions, we'll work on this um quadratic equation.

[00:03:41]How do you know this is quadratic?

[00:03:43]Because of the power of two.

[00:03:46]Okay?

[00:03:47]And because of the power of two, we are going to expect two solutions from there.

[00:03:52]So, let's take it up from there.

[00:03:57]Okay, so we are using quadratic um equation formula for this.

[00:04:02]The formula is x equals minus b plus or minus we have square root of b squared minus 4 a c everything is over two multiplied by a.

[00:04:21]So, what we'll do now is to know our a b c.

[00:04:24]a is the coefficient of x squared, which is one. So, a is one.

[00:04:29]b is the coefficient of x and that is two.

[00:04:35]So, our b is two.

[00:04:38]And then c is a coefficient um the constant, rather which is equal to four.

[00:04:45]So, we're going to put all of these into the the equation right away so that we have x to be equal to minus b, that will be minus two plus or minus b squared, that's going to be two squared.

[00:05:04]Then we have minus four times one, a is one, right?

[00:05:12]Then times four.

[00:05:14]Because we know that c is also four.

[00:05:19]And everything is divided by two times one.

[00:05:25]Divided by two times one, cuz our a is still one.

[00:05:31]Now, the value of x is minus two plus or minus the square root of two squared is four minus 16.

[00:05:42]Four times one times four is 16.

[00:05:45]And it is over two.

[00:05:48]Don't divide two yet. Don't let two divide yet.

[00:05:52]Okay, until we clear out the the um square root sign.

[00:05:59]So, from here now we have our x to be minus two plus or minus the square root of minus 12.

[00:06:10]Over two, cuz four minus 16 is minus 12.

[00:06:15]Then we can simplify this root negative 12.

[00:06:19]And we'll have minus two plus or minus square root of 12.

[00:06:25]Can see that the negative has not come down.

[00:06:29]So, we need to multiply this by square root of negative one.

[00:06:33]And everything is still over two.

[00:06:37]By the way, our x now is minus two plus or minus square root of four times three.

[00:06:44]Because there's a perfect square that is a factor of 12. So, four has to come out.

[00:06:52]Then multiply by square root of negative one, which is i, imaginary.

[00:06:58]This is all over two.

[00:07:01]Now, the next point the next point is that we break this point.

[00:07:08]You know, this can be broken to give us um square root of four times square root of three, right?

[00:07:16]So, meaning that x is minus two plus or minus square root of four there will be two, then multiply by square root of three.

[00:07:26]In fact, before we multiply mul- um before we multiply by the root, multiply two and i, that will give us 2 i.

[00:07:35]Then we multiply by root three.

[00:07:38]And this is over two.

[00:07:41]So, if you go on you can now divide the numerator by two.

[00:07:49]So, that's our x will now be minus two plus or minus we have um No, not minus two again. Two into minus two is minus one.

[00:08:00]Then two into 2 i will give us This is one.

[00:08:06]Two into 2 i will give us just i, then we have root three.

[00:08:11]But, this is plus or minus, so it's a two-in-one solution.

[00:08:15]Now, let's bring the three solutions together already.

[00:08:20]Okay, so before now, if you can remember we got x1 to be equal to two. This our first solution.

[00:08:29]Then x2 is from here minus one plus i root three. This our second solution.

[00:08:38]Then third solution is minus one minus i root three.

[00:08:45]So, these are the three solutions to the equation.

[00:08:49]Thank you for watching. Subscribe, like, comment, and share.