 So most of you would be wondering what is wrong with our Khilsa today? Why is he showing us the videos of cats and tigers? So, dear all, this is a small attempt from my side to bring out the distinction between the discriminant of a quadratic equation and the discriminant of a cubic equation. Does a cubic polynomial equation even have a discriminant hat? Friends here will do some flow tests, will ask people whether they are actually aware. I am very sure that many people are not aware of cubic equation discriminant at least. So let's find out. Okay, so first we'll start from Chaitanya. Chaitanya, have you heard of quadratic equation discriminant? No sir. Have you heard of cubic equation discriminant? No sir. Okay, let's ask Krishna now. Krishna, have you heard of quadratic polynomial equation discriminant? Have you heard of cubic equation discriminant? I know. Let's ask Pragun. So Pragun, have you heard of quadratic equation discriminant? No. Discriminant of a quadratic equation? No. Have you heard of discriminant of a cubic equation? No. Okay, let me ask Bharat. Bharat, have you heard of quadratic equation discriminant? Yeah. But have you heard of cubic equation discriminant? No. Seethu, have you heard of quadratic equation discriminant? Yeah, yes sir. Have you heard of cubic equation discriminant? No, don't have it. In the morning, sir, have you heard about the Discriminate of Quality Equation? No discrimination policy at Sentom. We tolerate all this. All are equal before us. In the afternoon, sir, have you heard about the Discriminate of Quality Equation? Discriminate of Quality Equation? Dharacharya had told me about it. Dharacharya is not Dharacharya. No, Dharacharya had told me about it. But have you heard of Cubic Equation Discriminate? Cubic Equation Discriminate? Oh my God. I don't know. Can you tell me? Is it real? Now, those who are wondering what's a discriminant, a discriminant is basically an expression whose value helps us to discriminate between the nature of roots of an equation. So, for a quadratic equation, let's discuss the discriminant briefly. So, let us quickly recap the quadratic equation discriminant. So, let's say we have a quadratic equation A x square plus B x plus C equal to 0. So, in this quadratic equation, I am assuming that the coefficients of the quadratic equation are all real. So, my A, B and C, they are all real numbers. And of course, we know that A cannot be 0 in a quadratic equation. All right. So, what are the discriminant in this case? The discriminant in this case is this term B square minus 4 AC because this term helps us to discriminate between the nature of the roots. How does it do that? Let's see that. If the discriminant is positive, then what does it mean that the roots of the quadratic equation are real and distinct? They are real and distinct. Okay. And if this discriminant is equal to 0, the roots of this quadratic equation are real and equal. And finally, if this discriminant is negative, then the roots of the quadratic equation are non-real or imaginary. And one more thing I would like to add on here is that if the coefficients of the quadratic equation are all real numbers, then these two imaginary roots will be conjugate pairs of each other. So, if one of them is of the form P plus IQ, the other would be of the form P minus IQ. I hope this information is known to everybody who's watching this video. Now, time for some scary part. So, talking about a cubic equation, let me assume my cubic equation to be AX cube plus 3 BX square plus 3 CX plus D equal to 0. Okay. Of course, here also I'm assuming that your coefficients A, B, C, D, they are all real numbers. Okay. So, A, B, C, D, they are all real numbers. Now, in this case also, there is a discriminant which is associated with two terms. First, I'll talk about a term called H. This H is basically given by AC minus B square. So, as you can see, the coefficient of X, which is 3 C, take out the C from there. The coefficient of X cube, which is A, take the A from there. And the coefficient of X square is 3 B, take the B from there and make this term H, which is AC minus B square. Okay. There's another term G, which is slightly complicated. It is given by A square D minus 3 A, B, C plus 2 B cube. So, here the term G is A square D minus 3 A, B, C plus 2 B cube. Okay. Now, actually we go for a determinant. So, the determinant here is given by D is equal to G square plus 4 H cube. Okay. So, this term G square plus 4 H cube. And the reason why I gave it separately as G and H is that if I would have written all of them together, it would have become a very huge term for anybody to remember. This D plays a major role in deciding the nature of the roots of the cubic equation. Now, first let us talk about what all types of roots can I have for a cubic equation. I can have four cases. I can have a case number one, where all the roots are real and distinct. Number two, where two roots are equal and one root is different. Case number three, where all the roots are real and equal. And case number four, where I have one real root and two roots imaginary. So, let us see how this discriminant D plays a major role in me deciding which of the scenarios will hold true. So, if this discriminant D is lesser than zero. In that case, you realize that this equation will have three real and distinct roots. Okay. So, the roots will be all real and of course, they will be all distinct from one another. Means no two roots will be the same. Okay. Next, if your D value is equal to zero and apart from that your GNH, they are also equal to zero. Okay. In this situation, you would realize that it would have three real and equal roots. Three real and equal roots. And mind you, my dear friends, if my D is equal to zero, but my GNH, they are not zero individually. That means my GNH, they are non-zero terms, but still my D term is equal to zero. Then in this case, I will have two real and equal roots. Okay. Plus one distinct real root. Okay. So, what am I going to get? I am going to get three real roots out of which two will be same roots. Means there will be a repetition of the roots and one root will be distinct from the other two. And coming to the last case, when your discriminant is positive. Okay. So, when your discriminant is positive, please note this down that you will have a case of one real root and two non-real root or you can say two imaginary roots. Okay. So, one of the roots will be real. That means the x-axis is going to be cut by the cubic polynomial function only at one point. And two of the roots will be imaginary in nature. And again, I would like to repeat here that if the coefficients of the cubic polynomial equation are all real, these imaginary roots would be conjugate pairs of one another. So, hi-haul, I will be just showing you the demonstration of how does the discriminant basically decide the nature of the roots. So, let's have a cubic equation. Y is equal to A x cube plus 3 B x square plus 3 C x plus D. Okay. Now, you can see here that since I have not chosen the value of A, B, C and D, the system is smart enough to basically pick up some values of A, B, C and D because it treats them as parameters. And right now it has kept them as one, one, one each. Okay. However, it has given us a liberty to change these values from minus 5 to 5. Okay. Now, here is a situation where the tool has actually taken up A, B, C, D each as one. Okay. So, let us find out H first. So, H is A, C minus B square. Okay. So, A, C minus B square as you can see here is coming out to be zero. And G value is A square D minus 3 A, B, C plus 2 B cube. Okay. And the discriminant here which is G square plus 4 H cube. Okay. So, right now here you see a scenario where H is zero, G is zero and D is zero. So, this is basically a case if you recall from the flow chart. It's a case where the discriminant is zero along with the fact that H and G, they are also zero each. So, in this case, the cubic equation will have three real and equal roots which you can see right on your graph over here. The three real and equal roots here are minus one. So, the equation, the cubic equation has got three real equal roots which is minus one, minus one, minus one. Okay. Now, what I'm going to do is I'm just going to vary this value of C slightly. And I'm going to vary it in such a way that it is only cutting the graph at one point. So, this is a scenario where you see that the function has only one real root and two imaginary root. So, this is a situation where you see the D value to be positive. So, you can see down here I'm showing it with my cursor 3.038. Now, I'll be shifting the graph a bit down so that it has got three real and unequal roots. So, this is a situation where you'll see the D value becomes negative 1.332. So, just quickly recapping it, the value of D here is negative which basically ensures that the cubic polynomial equation has got three real and unequal roots. Now, I will slowly shift the value of D so that the function touches the x-axis at one point and cuts at another. So, basically this is to ensure that the function has got two real and equal roots and one root is distinct. Okay? So, I'm trying my best to best take it up and you can see here that the D value is almost zero. It's very difficult to make it zero because you need an accurate positioning of the graph. So, here is a situation where h and g, they are non-zero values but D is coming out to be zero which was as expected in the algorithm. So, I hope the discriminant of cubic equation did not scare you that much. But you know what? Let me give you a piece of respite here. Let me help you to tame this tiger by giving you a small mnemonic over here. Alright, so the memory mnemonic goes like this. So, we have to first remember this fact that D is g square plus 4h cube. Okay? So, you know, you can make your own mnemonic for remembering this but this is something which I would request you to remember on your own. Yes, for remembering, h I would give you a small suggestion. Okay, so normally what I follow is h for home, a for ac. Okay, ac is your air condition minus b square. b square I'm treating as two b's. So, big bills. So, I'm reading this as home ac without, minus is without, big bills. Right? So, this is the way to remember h. Now, coming to g, how do you remember g? g is basically a big expression which is a square d minus 3abc plus 2b cube. Okay, so this is how I break this up. So, I break this up as a square d as aad minus 3abc plus 2bbb. Okay? So, I've just written this expression down. So, this is how I make a mnemonic out of it. So, this is basically army attempts to destroy. So, army attempts to destroy aad, three Afghani bomb cars. So, army attempts to destroy three Afghani bomb cars, but two big bombs still burst. Okay? So, again a very funny way of remembering it, but this is what you can actually, you know, remember if you want to. So, army attempts to destroy three Afghani bomb cars, but two big bombs burst. But means they wanted to destroy three, so minus three, but they could not. So, plus two, big bombs burst. Hey, I hope that Ramani gave you some sigh of relief, didn't it? And it will also help you to remember the formula for the rest of your life. Because you're going to see the application of cubic equations in future part of your studies during engineering. Thank you so much for watching this short video. Bye-bye. Take care and stay safe.