Cardano's formula_part.3 11.3.2026 | Claire & Bruce

Added: 2026-05-10

[00:00:01]Hello, I'm Claire. Today, let's talk about the third video of the Cardano equation. So, in this video, I'm going to add everything that about equation that I didn't tell you before.

[00:00:14]So, um so the first point we know that uh for example, we set it set a question as x3 - 3x um + 2. So the question is over here then we know that we set in this question using u + v or as the hook function right m + k / m and if it's u plus v so we know we can just think about that u and v we change their root right we change their root and then so in the three dimension I talked about it before also so in this video I'm going to talk about it actually because we know because in this question right now we will find out the u is equal to square root so in this question I'll tell you in this question we find this question when delta equals z and so here's the point when delta equals z okay then we will find out um because it's a cube root right and then the inside is negative 2 over two. So, so that is -1 right in this case because it's okay 2 / two that's -1 over here you're going to plus square root delta so I told you this is equal to zero of course you're going to als Oh no.

[00:02:00]How about V?

[00:02:03]Okay. V is equal to square root is sorry cube root and then um -1 minus square<unk> delta. Then of course this is equal to1 and this also=1. So x = -1 + -1 = -2. Great. So this is the normal method. But actually we have another setting method is that we set x equals the u + v. So that's u minus v. But when we think about we let u and negative v change their value change their root it's u axis and negative vaxis not u and vxis then we cannot change because the symbol over here it is negative but the correct thinking method is think about you negatively these two axis okay so what do we have Um let's take a look. Um so so in the processing it's totally same but the difference it's just U and B. So actually finally we're going to find a U and B. Actually going to find our negative B but we actually over here we just write it as B. So we have square root sorry cube root and then -1 -2 /2 and then plus square<unk> delta okay then in this case it is different then v it is equal to cube root and over here it is positive q /2 so that is one and it's the same simple over here but this method the the two symbols are opposite in these two symbols are the same over here and then square root delta. So in these so in this case then u is equal to1 and v is equal to 1 and so this is correct because in this time in this second we'll have -1 +1 because it is think about negative v over here so that is correct okay so that is also equal to -2 so we're going to compare these two methods are their answer the So of course we know these two symbols can change and these two symbols they can change both change to negative. So let me write out the first is that x= u + v.

[00:04:50]Okay. And then this is equal to square root sorry cube root and q / 2 then plus square t delta and then um plus cube root q over 2 minus square<unk> delta and then or square root sorry cube root and then there's the same Q over two but this is minus square root and plus cube root Q / 2 + square<unk> delta x = u + negative v. Okay, in this case and that's equal to square root sorry cube root and then Q over 2 plus delta and then plus cube root Q / 2 q over two and then plus square root delta and then or equal to or square root sorry cube root and Kative - Q / 2 and minus square<unk> delta sorry you're supposed to be minus minus square root there's cube root Q over two minus square root delta. Okay, so we're going to compare. So, first compare. Of course, these two guys, of course, they're the same.

[00:07:14]I'm going to think about these two guys actually are same because if you put on put in a negative over here then that's over here it's negative Q /2 minus square delta and over here this is negative Q /2 then plus square root the delta. So these two actually they're also the same. Okay. So we can just compare because for example can compare these this guy with this guy and this guy this guy. Okay, we compare if it's this with this and these two.

[00:07:54]Um, not like this supposed to be.

[00:08:14]Okay, these two guys, these part these two parts are same. So we put the ne negative symbol over the inside to the inside. It's also same.

[00:08:27]See this same if these two parts same and put this negative inside negative symbol inside. Okay, then these two guys are same.

[00:08:41]Okay. So actually now we have done is to compare x plus x= u + v and x= u plus negative v. They're actually the same.

[00:08:53]So the first method mapper map mapped to the second method the difference is v one is v and another one is ne. So I told you what have we done every step we actually from the first method second method just change V to negative V remember to plus a bracket and so when we are going to find the final root of uppercase U and uppercase V is that in this case we're going to actually going to find the root of negative B right so actually here I should have just supposed to write plus negative bracket bracket it negative bracket. So in this case we know how to do this. Um okay so this is the first thing um we know that we can use u plus negative v. It's totally same because I told you we change v to negative v. So when you graph we think about the negative v axis so nothing changes.

[00:10:00]Okay.

[00:10:04]Um so the next step is tell you another method.

[00:10:10]Um okay. So the next thing I'm going to tell you is about the another method.

[00:10:25]So the first thing we're going to know um is about a minus b. You feel a bit strange about this. Okay. So first step is to change it to square<unk> a plus square<unk> b time square<unk> a minus square root b.

[00:10:51]You know what's that right? Okay.

[00:10:55]And the next step now we can change it to bracket um bracket cube root a then minus uh cube root b doing a times about the cube root and then that's um brack um no no bracket cube cube root a squared because it's cube root a * cube root a okay because it's actually this whole thing t* times three times it and when we times two of it and just it's plus square root sorry cube root a b actually it's cube root a time cube root b so we're going to put them together that's okay and then plus cube root root B2.

[00:12:02]Okay.

[00:12:03]So, next step is about we think about a plus b.

[00:12:12]So that is gonna equal to square root a sorry cube root a and plus cube root b okay so then times okay and a square cube root a square minus uh cube root a b plus cube root b².

[00:12:51]Okay.

[00:12:53]So here over here now what we know over here we can just solve the question. So question is x to the 3 + 9x - 10. We're going to find out when this room. Okay.

[00:13:09]So here's a new method. Then yes. So x 3 + 9 x - 10 and this is equal to x to the 3 - 1 + 9 x - 9. So we're going to be having uh x - 1 bracket x to the 2 + x + one then plus 9 * bracket x - 1. Great. So of course then that's equal to bracket x - 1 * x to 2 + x + 10.

[00:14:01]Good. Then we will know we can make we can guess of this function and x -1 * um x + 1 / 2 bracket 32 + 39 over 4 Great.

[00:14:34]Okay. So, what is this equal to? So, I mean know what this we're going to find with zero. So, this part equals all this part. But this um quadrant equation did not have root. I mean real number root of course. So, we only can find out that we let x -1 equal zero. Then x = 1. This is one root that we can actually find out. And then we know using the Cardano method. Um I don't write the um I just write a result and just use these three uh equal 5 + uh 2 square<unk> 13.

[00:15:27]And then these 3 to these three is equal to 5 - 2<unk>3.

[00:15:36]Great. When we have this then we know x is going to equal to u + v and that's equal to cube root 5 + 2 square<unk> 13 and plus square<unk> 3 square<unk> 3 cube root and then 5 - 2 time square to 10, right?

[00:16:08]When we have this, we're going to find out. Oh, so we have a square root thing.

[00:16:15]So, how can we know what's the real number of it? So actually this we can use other tricks to find out a value that we know for example actually answer of this I'll tell you it is one but we can use trick to find out this so first we know x is equal to this okay so post So x to the 3 and that's equal to 5 + 2<unk> 13 and + 5 - 2 *<unk> 13.

[00:17:08]Okay.

[00:17:11]Then then we're gonna then we're going to plus 3 * square root sorry cube root 5 + 2 *<unk> 13 and then time cube root 5 - 2 * 13.

[00:17:39]Of course. Then we're also going to times square root sorry cube<unk> 5 + 2 square 13 and plus cube<unk> 5 - 2<unk>3.

[00:18:01]Great.

[00:18:03]What is it equal?

[00:18:07]equal to over here of course the 10 plus.

[00:18:14]Now this part turns to three times square root sorry cube root 25 uh - 4 * 13 right okay so this is not this way this Okay.

[00:18:45]So this case we'll find out that x is equal to not no not over here times x.

[00:19:04]Okay. So we'll find out this equal to 10 * 3 * cube<unk> -27 * x. Then that's equal to 10 - 9 x.

[00:19:22]Okay. Then x3 + 9 x - 10 = z.

[00:19:32]Then we will have x - one. Okay. Then x²ar + x + 10 = zero.

[00:19:46]Then we have x² x I don't write it because this part we don't have a root and so x= one. Okay.

[00:19:59]So this is it. Then we'll have that this whole part is equal to one.

[00:20:08]Yeah, you get it right.

[00:20:11]Okay.

[00:20:13]So now over here is a point because we actually can think this part as of course including the cube root. So cube root. So inside we can write it as m + n square root k.

[00:20:38]So there's some conditions and the condition first condition is n greater than k and a is going to be equal to this is what we set as a and then n equal to n minus k over 3.

[00:20:57]Great. And a uh a to the 3 + 3 a k equals m. So what is this equal to?

[00:21:13]Wait.

[00:21:21]So what is this equal?

[00:21:24]So um we'll think if we have a equals this guy and then so a d3 3 * a to the 3 + k is equal to n right.

[00:21:41]So in this case including this if we put it inside then we will actually have inside the cube root it turns to bracket a + square<unk> k bracket 33 and so that is equal to a + square root Okay.

[00:22:12]So over here is that we will have some trick else.

[00:22:22]Um so let me see.

[00:22:27]Okay. So the another trick over here is also about this part. So first we will have um no I mean this part. So first we know that so inside so five + 23.

[00:22:55]So we can just change it to we can just we can just change it to to five + 4 square<unk> 13 over 4. Great.

[00:23:12]Okay. M over here is equal to five and n = to 4 and k = to 13 over 4.

[00:23:23]So this cannot be true because two is less than 13. So we cannot use b because n has to greater than k. So over here in this case then it satisfies. Okay.

[00:23:36]So then what is a equal to? A is equal to that. Okay. Then n is equal to four, right? And so it's square root 4 - 13 over 4 then whole thing divides three.

[00:24:02]Okay.

[00:24:04]What is this equal to? So this is equal to square<unk> um 16 - 13 / 3 * 4 and so that is equal to square<unk> 3 over 3 * 4. So it's equal to square<unk> 1 / 4 and then that's equal to 1 /2.

[00:24:31]Okay. So what about this guy? We find a and a to these three plus 2 2 a k. So a to these three it is one over eight.

[00:24:44]Okay. So it's 1 over 8 then plus 3 * so this is 1 / two right and times k over here is this guy. Okay. And then so we have 3 * 13 over 8.

[00:25:04]So it is equal to um 1 + 39 over 8 and then that's equal to 40 over 8 and that's equal to 5 = m. So that's correct.

[00:25:19]Okay. So the next step because we know what's inside over here and so the whole thing um three So sorry the cube root the uh cube root and then so we know the whole thing is one over two one over two we have this and then plus square root 13 over 4.

[00:25:52]Okay, the whole thing to these three.

[00:26:01]Okay. So, so we actually have that. Okay. So, 1 / 2 +<unk> 13 over 4.

[00:26:14]Okay. So, okay. We have another part, right? and another part because I'm using what we know over here because their sum have I mean their sum has to be one then we know uh cube root 5 - 2<unk>3 has to equal has it should be equal equal to what? It should be equal to 1 / 2 -<unk> 13 over 4.

[00:27:00]So when they close together, that's what I want. Okay. So here's something I'm going to add. So um because I just lost the um you know, I'm just going to add another um example. And here in this example it is when delta is us and zero.

[00:27:16]That's why I have this picture right. Um so this question is let's see um this question is that um x to the 3 - 7 x - 6 and we're going to let it equal zero. So actually we can actually change it too.

[00:27:38]So if you know this problem a lot, you'll find out that you can just change it um x is three um + one then minus um 7 x - 7. So in this case you'll have in this case you'll have that is equal to um x + 1 and then time x to the 3 - x - 6. Great.

[00:28:08]And you'll have you'll have um x + 1 * x + 2 * x - 3 and that's equal to zero. So you get the answer is x1 = -1 x2 so just x1 and x2 = -2 and x3 = 3. So this is the point. So if we had these three roots and another thing if you use goddamn equation you'll find out.

[00:28:47]So first the point is here if we use the canon equation because in the last video I talked about it we use the canon equation when this is lesson we have a special method to calculate it. So in this case we calculate it using that special method because in that special method it can just use a special method and calculate and find the three roots that they are all real number root. It's not like just if you just use a normal U plus V and omega U plus omega 2 V in that case you'll find out that you exist I so don't know how to how to solve it so maybe you you know you'll you'll just the I will vanish but you don't know how to prove it or some kind of you know then we have a special method then in that case we use the Cardano equation we have we will have x1 is = 3 and x2 is = 2 and I'm sorry -2 and x and x3 is equal to1. So in this case we have these readers using using continental equation this um kind of problem in this kind of problem. So in in this po in a point over here is that if we use if we use a normal if we use a normal equation you know we have the u plus v like I said you'll find on the x1 is equal to u plus v right and u plus v is equal to cube root um 3 + 10 i over 3 square t over uh one over.

[00:30:37]Okay. And then plus cube root three. That's not cube root just just cube root. Um and then 3 - 10 I over three then square<unk> 1 / 3. So this case we had these two guys. And what are you going to do? We have two tricks. We have two tricks. And as I told you before in this video in in in that black whiteboard I just told you you know I just say that we know we can create a trick. We can create a trick. So if we have some kind of uh square root m + m * square k it's sorry this is cube root and in this case we can change it to uh sorry cube root a + um bracket a +<unk> k and then bracket to be three and that's equal to so that's equal to a + square t right in this case a is is is equal to square root n minus k over 3 and and a squar uh sorry a to the 3 + 3 a k is equal to m.

[00:31:55]So because satisfying these twos I will have you know this equal to this. So in this case I had this trick this is a trick.

[00:32:08]So when I have this trick I can I can finish this problem when I had this. So I can change I I I can I I can change this thing looks like this. Actually it's already like this but we can just change it any more. Okay. So So the first step is because we have a eye. So we're going to just put because here are just all some real name but with no eye. So we're going to just put eye back in.

[00:32:43]Right. Plus um okay got this cube root right? Um so in this case so we're going to actually change it.

[00:33:02]So um actually we can just pick a negative out of here. So it's equal to negative okay and then square this way plus so this negative right and okay and then minus cube root 3 10 square root 3 great and in this case we can change it to we just put a negative inside here and so of course here we had negative and that is cube root so it's -3 and we're going to minus what do we do put the two in this so this begin with four and this is five or three so five over 3 * square<unk> 4 over 3 great and then we plot again right and it's the same but just going to here sorry okay So plus one put in three and then plus 10 over three.

[00:34:13]Sorry here's five.

[00:34:15]So 4 -4 over 3. Great bracket. Okay. So now in this case we know m is equal to -3 and n is= 5 over3. Then k is equal to -4 over 3. And if you calculate it, you'll find out that a is equal to one.

[00:34:38]And so in this case, if you plug this in over here, it does satisfy that these satisfy this condition. It's one m. So because actually when you change this, you're just going to find out that these this actually satisfy this this this condition, right? other in this case you can just calculate it find it you just need to plug it in but this situation so what do you need to find to satisfy because in this case you just because of this you already in short a in this case you have to plug it in and let it satisfy this what we finding for so a= 1 so a= 1 then you if you plug it in you'll find out that so this is equal to so u equ= Um, yeah, it's u u = 1 + 2<unk> 1 / 3 and v is equal to 1 - 2<unk> 1 / 3. So in this case we have these two guys. So if you calculate it so you have x1 is equal to -2 and x2 is equal to -1 x3 = to three. So in this case this is very different you see in the cinal equation right we'll find out the first root is three. So in in this case the first root it is -2. So we have another method that we can calculate out. We can find the the root the the root of the root the first root were three were three. Okay it is three.

[00:36:27]So the point let me tell you this first then I'll say the end. Okay. Say the odd word. Okay. Um so in this case we will have what? Because we have this root right. Okay. And in the mapping to this this picture we know that means x1 in this case because this only mapped to the c. So in the c. So it's this this is three this two and this is one.

[00:36:54]So I'm kind of like I I say this in the last video. I just think of I just draw this. Okay. This is uh this is three and -2 and1. So in this case I can use this trick to find out this three and -2.

[00:37:08]This is just, you know, I can just, it looks like a pair, but you couldn't just say these two guys were wrong because we could go as and this. Okay, let me just use the other part. Um, so let's see. Um, it's actually very strange because we use we use the same equation and we use a different trick.

[00:37:31]We can change to a different answer. we can use some some uh higher thing. I don't know how to prove it now but in instead now we know because it actually exists. Um so here we have another trick.

[00:37:51]So this trick is so first of course we have we have we have this. Okay. And we're also going to put it in. So this step is the same.

[00:38:20]Okay. Um so in this case what's there's a trick. It is actually equal to square root oh sorry cube root um 3 + 20 over 3 and then square root um 12. So we can just here's a two. So here we need the one over four. So right because we put in the square root. Okay.

[00:38:48]So then plus three 3 - 20 over 3 square<unk> 1 over 1 / 12.

[00:39:06]Okay. So in this case we have you know we can just map to this again.

[00:39:17]And so we have m is = 3 and n is = 20 over 3 and then k is equal to -12 sorry 1 over 12.

[00:39:32]Okay. So in this case um in this case we also satisfy that we can find out a is a is equal to uh 3 over2. So in this case we can actually find find a also if we just plug it in we also if we plug it in also satisfy square okay so in this case you'll find out that um x1 x1 is equal to um uh 3 and x2 is equal to -2 and x3 is equal to 1. So in this case we use this trick actual fun oh just like this. So the point over here is that we can use the trick to find out these two kind of roots. Okay. So this is what I'm going to add. Okay. And let's end this video.

[00:40:44]Oh, of course I'm going to tell you. So, next video I'm going to I'm going to talk about the shenzing formula. Um, it's about we just changed a little bit from the uh from the card equation, but it's a very, you know, uh, it's very much I have to write just a lot of calculator. You just opposite thinking I'm just going to create a formula. So, I just, you know, okay, so this is what I'm going to talk about in next video.