Olympiad Mathematics | Indian | Can You Solve This?

Añadido: 2026-05-11

[00:00:01]Hi, everyone.

[00:00:04]If you're ready, let's provide a solution to this problem here.

[00:00:11]Solution.

[00:00:15]Okay, we have x + 2 to the power of 4 equals 81.

[00:00:24]How do we solve this problem here?

[00:00:28]Let's work it together very quickly.

[00:00:31]This is x + 2 to the power of 4.

[00:00:37]Right? So, the first thing we'll do is to get the square root of that.

[00:00:44]Then we have plus or minus the square root of 81.

[00:00:50]Now, this one will go into this two times.

[00:00:55]So, that we can have we can have x + 2 to the power of 2 to be equal to plus or minus square root of 81, and that is 9.

[00:01:11]Okay, so what do we do at this point?

[00:01:14]Let's expand what we have on the left.

[00:01:18]Remember that if you have a + b to power 2 this is the same as a squared plus 2ab plus b squared.

[00:01:36]Okay, so that is the expansion.

[00:01:39]And on the left-hand side, the expansion now will give us x squared plus 2ab, that will be 2 * x * 2, and it's 4x.

[00:01:52]2 * 4, okay, 2 * x, then * 2.

[00:01:58]That will 4x because our A is um x and our B is two.

[00:02:04]Then we have plus b squared, which is two squared. Two squared will give four.

[00:02:10]So, the expansion here is this.

[00:02:13]And this is equal to plus or minus nine.

[00:02:17]Now, this means that we're going to have two um equations because of this plus or minus.

[00:02:25]One is going to be positive and the other will be negative.

[00:02:28]Now, let's work with the positive one first.

[00:02:32]So, the equation will now be x squared plus 4x plus four equals nine.

[00:02:42]So, bring this to the left as we have x squared plus 4x plus four minus nine equals zero.

[00:02:54]Okay, we're trying to get our first um quadratic equation.

[00:02:58]So, x squared plus 4x minus five is equal to zero.

[00:03:08]So, now we have to look at this um look at this equation and see if we can factorize it. This is a quadratic equation because of the square here because of the variable x and the equal sign.

[00:03:27]The square, the variable, and the equal sign.

[00:03:30]>> [snorts] >> Okay, so these are the conditions that make an equation to be a quadratic equation.

[00:03:38]Now, what method we use to solve it?

[00:03:41]Let's look at it this way if it's what we can factorize.

[00:03:46]What are the two numbers we can multiply to get minus five?

[00:03:50]And then we add the numbers to get + 4.

[00:03:55]Okay, I think they are The numbers are going to be um The numbers are going to be five.

[00:04:04]They're going to be five and minus one.

[00:04:08]Yes.

[00:04:09]The numbers are going to be five and minus one because 5 * -1 is -5. Then 5 - 1 that will give That will give us + 4.

[00:04:22]So, we are good. The factors now will be x + 5 into x minus five.

[00:04:32]And this is all equal to zero.

[00:04:36]Now, we are multiplying this and this together zero, so we can apply Oh, by the way, this is supposed to be one. Okay, so this is minus one. So, the product of the two is giving zero, so we can apply zero product rule.

[00:04:51]Let's do that.

[00:04:54]Okay, so we will apply our zero product rule to this so that x plus five is either equal to zero or x minus one is equal to zero.

[00:05:09]From here, x will be 0 - 5 or x will be zero plus one.

[00:05:19]So, here now we have x to be equal to minus five.

[00:05:24]Or here our x is equal to one.

[00:05:28]Right? So, this is um the solution. Let's conclude from here.

[00:05:35]Our x is equal to minus five or plus one. So, here we have two solutions.

[00:05:43]Now, we need to get two more solutions and that will come from the other equation that we left out.

[00:05:50]It is x squared plus 4x plus 4 equals minus 9.

[00:05:59]Okay, the one we are using we solved to get this.

[00:06:03]We equated it to plus 9, so now we have minus 9.

[00:06:07]Now, bring everything to the left. We have x squared plus 4x plus 4 plus 9 equals 0.

[00:06:19]So, to continue with this we'll have our x squared plus 4x plus 4 plus 9 is 13.

[00:06:30]This is equal to 0.

[00:06:33]Okay, so what again do we do? We're going to solve this using um our quadratic formula method because we cannot factorize this.

[00:06:43]And x is minus b plus minus we have b squared minus 4ac.

[00:06:51]This is over 2 * a.

[00:06:54]A is 1, b is 4, c is 13.

[00:06:58]So, [crying] let's put all of them into this formula right away.

[00:07:04]So, that our x will be place of minus b we're writing minus 4.

[00:07:09]Plus or minus b squared is going to be 4 squared.

[00:07:14]Then we have minus 4 * 1 * 13.

[00:07:19]4 * 1 * 13 because a is 1 and our c is 13.

[00:07:26]This is going to be over 2 * 1, which will still give us 2.

[00:07:31]So, at this point we have to simplify what we have under the root.

[00:07:37]So, x is minus 4 plus or minus 4 squared is 16 minus 4 * 13 is 52.

[00:07:48]52 and it's over two.

[00:07:51]Yeah, that is over two. So, from here we are going to continue.

[00:07:59]Okay, so we are going to have X to be equal to minus four plus or minus square root of minus 36.

[00:08:12]That is 16 minus 52.

[00:08:16]And this is all over two.

[00:08:19]Now, what do we do?

[00:08:22]To find the square root of negative 36, we're going to get 36i because of this negative attached to it.

[00:08:30]So, our X will now be minus four plus or minus um square root of negative 36, like I said, is going to be 6i.

[00:08:41]And this is all over two.

[00:08:45]Mind you, what we have there is a two-in-one solution because of the plus or minus.

[00:08:50]Let's continue.

[00:08:53]Okay, so this means that X is um two into minus four, that is minus two plus or minus two into 6i, that's going to be 3i.

[00:09:06]And this is a two-in-one solution.

[00:09:09]Now, to bring the complete solutions together we got X to be equal to minus five. This is our first solution.

[00:09:19]Then we got X again to be equal to one, our second solution.

[00:09:25]And we are having X from here to be minus two plus 3i our second our third solution.

[00:09:37]Then X to be equal to minus two minus three I and this is our fourth solution.

[00:09:46]Thank you for watching.

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