NDA-II 2026 Maths LIVE | Quadratic Equation | Class-06 | Complete Theory + MCQs

[00:03:29]Hello hello no sound yet one one no problem turn it off the rest of the children turn on the mic the mic was on I do n't know how it got turned off okay let's see the same question again from the beginning the question is saying that if one of the root of the quadratic equation ax² + bx + c sometimes it gets done in a hurry. I have just come running quickly. He did not notice that there was no sound. I did n't even check the message. Ok? If one of the roots of the quadratic equation ax² + bx + c = 0 is equal to the nth power of the other root.

[00:04:26]Here we are talking about the nth power.

[00:04:28]I think, among all the questions I had asked, we had not asked any question of nth power yet. 1 / Alpha Alpha had seen it all. This was left over from your nth power of the other root times the value of ac power n to the power 1/n + 1 + a power n to the power c 1/n + 1 equal to your root here. The first root will be alpha. The second root will be alpha to the power n because that's saying n times the nth power times the first root.

[00:04:59]Ok? So these are your routes.

[00:05:01]What do you have to do now? knows two routes. If you know the roots, the first thing you will do is find its sum. We found one, alpha + apha power n = b / - b / a, so we left equation one here. Then Alpha* Voice has arrived.

[00:05:28]Alpha * Alpha to the power n extracted. What is this? This product is c/a alpha * alpha power n c/a so from here if we add the power to it then alpha power n + 1 = c / a and from here if we take out the value of alpha then the value of alpha will come out to be c / a power 1 / n + 1 this is the second equation. We take this value of alpha and put it into equation one. If we have to bring this relation. This is to bring value to this relation.

[00:05:56]So we're going to take this alpha value and put it into equation one. What will happen? What do you get by solving from equation from equation one and one and two? Put the value of alpha in it. The value of alpha is given to you c/a c/a to the power of 1/n + 1 up to a to the power of 1/ n + 1 plus alpha to the power n alpha to the power n is already there. Here c/ p/a was to the power 1/n. Here if n is there then it will become n / n + 1. will become n / n + 1. Equal to -b / a.

[00:06:37]= -b / a. Now we will multiply this a by both. We will multiply a from here. Ok? We will multiply a here.

[00:06:48]What will happen if we multiply? This a your this one a came. Here a to the power of a is 1 / n + 1. What happens when we move this a to the numerator? This will be minus power so a to the power of -1/n+1.

[00:07:04]Is it clear?

[00:07:07]into c to the power 1 / n + 1. We'll do the same thing here. Plus a * a to the power of -n / n + 1 - n / n + 1 and c to the power of n / n + 1 will equal to - b equal to -b. It is clear then when we solve this value, what will be our answer when we solve this value. Here the power of 1 is 1 - 1 / n + n + 1, so if we solve it, we will get a to the power of n / n + 1 * c 1 / n + 1. Here when we solve it, we will get a to the power of 1 / n + 1 * c n / n + 1 = -b = -v. Is it okay? Did everyone hear the sound now? Ok?

[00:08:09]This is your equation. The minus came.

[00:08:12]What did we have to bring? We had to bring a c to the power of n to the power of 1 / n + 1 + a to the power of n * c to the power of 1 / n + 1. Here a to the power of a * c to the power of n. So change this term here or write it like this. When you look here, you will see that it is written here a to the power n * c to the power 2 c to the power 1 / n + 1 I stopped a to the power n here. a to the power n stopped right here * c and here 1 / n + 1 was common in both. If there is power with both a and c then there is whole power with both.

[00:08:48]Plus a to the power of a here is 1 / n + 1. And c to the power n goes over here.

[00:08:55]What do we even have to bring? c to the power n.

[00:08:57]So we will stop it here. a c to the power n to the power 1 / n + 1 = - b I think this is the answer that was asked. I asked you about its value. What will be its value?

[00:09:09]Minus B Answer Option Number B.

[00:09:13]Ok? If the voice is not coming then please tell me. Is everyone hearing the voice? Is it clitty? What was the question asking? How did the answer come? If anyone has any problem, I can get it solved on the new slide.

[00:09:26]If you are facing any problem here.

[00:09:34]Move ahead.

[00:09:37]Now when nth root questions come, you guys will solve them. I hope you will do it.

[00:09:41]Alpha was the n root of the power of alpha. You first created an equation from the sum and then you found the value of alpha from the product. Put the value of alpha in it.

[00:09:51]After that simply calculate.

[00:09:54]Is it clear? Let's look further.

[00:09:57]If the roots of the equation 1 / x + p + 1 x + q = 1/ r equal in magnitude but opposite in sign. There are two things here. The first is equal in magnitude. The second is opposite in sign. When I explained the sign of the sign of the roots in the second class, I told you the conditions of both there. Your opposite in sign. Opposite in sign only yours which is also saying further here that give the product. Opposite in sign was the only case where your product was less than 0. c/a What was there was only and only opposite in sign less than 0. Everything else was positive. I am talking about c here. And equal in magnitude magnitude means we will take the value 3 -3. What is its opposite in sign and magnitude? It is equal.

[00:10:55]Ok? The magnitude is equal. Then the question is saying what will be the product of the roots product?

[00:11:02]Talking about the product. So if we look here, the roots of the equation are equal. You can take alpha. The roots will be equal in magnitude and opposite in sign. Correct? So let us consider this.

[00:11:15]It has roots. If we talk about alpha + beta. First, for alpha + beta, we will create it in quadratic form.

[00:11:22]I need to put this equation into quadratic form. So how will you make it? This is not μ. This is r, you could have solved it by taking r mu also. But many children have problems. So this is r.

[00:11:37]Solve this equation first. We will solve it here. x + p + 1 / x + q = 1 / r Solve this. x + q + x + p = Then this is also multiplied directly by r.

[00:11:55]Here I also got it multiplied by r.

[00:11:58]What will be the right hand side here? The value we get is x + p * x + q. We will solve this.

[00:12:06]Here x x becomes 2x. 2x + qr + pr is correct? than x² + x + xp + p okay? Change it to quad.

[00:12:27]Change to a quadratic equation. The first x² term will be the term containing the first x².

[00:12:33]After that, all the terms of x are brought to one side. Let's move all the terms of x to one side. What will happen? 2x here is your x turn xq and here is xp so both of these are in plus because we are taking x² on the right side. If we take these terms to the left side then x² here x common will be p + q and if we take it from here then x common will be 2r.

[00:12:59]What will be the final value? p + q and this 2x will come here so -2r 2x 2x give this value QR and PR and PQ these three are your constants. Then plus PQ - PR - QR this becomes your quadratic equation = 0 Now here you will see that the quadratic equation is being formed. So when you have to find the roots, what will be the roots? Alpha is alpha. So what will happen here when we calculate alpha + beta? Alpha Plus - Alpha Equ What will happen here? 0 will cancel out.

[00:13:42]And what is this? -b / a This is -b / a From here if we see the coefficient of b then the coefficient of b will be 0 sorry b coefficient is zero.

[00:13:54]Here b is the quotient of x. The quotient of x is b. So from here if we find the value of p + q - 2r then it will be 0.

[00:14:04]The value of p + q - 2r will be 0. Remove r from here.

[00:14:09]2r = p + q and the value of r becomes p + q.

[00:14:17]What is this value? of r.

[00:14:20]I got the value of r. Now what is it asking us to do with this than it is the product of the roots. Now let's talk about the product. When talking about product, a root is your alpha. One is Alpha.

[00:14:33]What will happen if we multiply this? What was the C/A equation? This is your equation. This is your C pq - PR - QR here the value of C is PQ - PR - QR we will write this we will write this PQ - R and P+ and what is the value of A? The value of A here is one. Ok? Is everyone's voice clear?

[00:15:10]Ok? There is value here. And the value of alpha* - alpha - alpha² = PQ - R was your P + Q / 2i. Here q p + q is already there. Once p + q is 2 this is the value we are getting. Take the LCM and you get 2 1/2. You will get 2 pq - p + q squared then from here take the minus common from the bracket.

[00:15:42]-1/2 p + q ho squared minus 2pq okay. Now that you know the whole square of p + q, what can we write for it?

[00:15:58]You know this. This is what I told you in the Roots form, this is what you had? -1/2 p squared plus q squared.

[00:16:07]This is your product.

[00:16:11]Ok? -1 / 2 p square plus q Now we will see which option is correct. Minus p squared plus q² / 2 is clear? If anyone has any problem, please tell me.

[00:16:32]Is it clear? First of all solve the equation.

[00:16:35]Solve the equation and bring it to quadratic form. Brought it in quadratic form.

[00:16:39]You were saying here that the magnitude is equal. And is the opposite sign.

[00:16:43]So three alpha and - alpha we took the route. When you calculate its sum, the sum will be zero.

[00:16:48]- b / a will come out to be 0. The value of b is zero. Which was the coefficient of b in this equation.

[00:16:57]With the help of p + q - 2r, which was the coefficient of b in this equation, I found the value of r here. After finding the value of r, I calculated its product.

[00:17:06]Product - alpha² pq - r p + q Put the value of r here. Simplified it after putting the value of r.

[00:17:14]This is becoming our answer.

[00:17:18]Ok? Move ahead.

[00:17:21]Next if 1 / 2 1 / 2 - -2 is one of the roots of the equation ax² + bx + c = 0 where a b c are real then what are the values ​​of a b exactly. Here you are being asked the value of A B C. Till now you were asked the value of A B C in a question where I told you about the series.

[00:17:42]APHP's question is asking A B C.

[00:17:45]What is the plan? First of all, what is your first route? 1/2 - -2 Simplify this first. If you talk about the first route. What is your first route? 1/2 - -2 here is the minus inside the under root. Going to the complex. will become i². The value of i² is -1.

[00:18:08]So here we can put i² in place of minus.

[00:18:11]Keep i². 1 / 2 - i² * 2 i² will come out to i. The root will be removed.

[00:18:21]1 / 2 - i 2 What to do now?

[00:18:26]To simplify, multiply by the complex conjugate.

[00:18:28]top down 2 + i / 2 + i up so your beans will be saved. What's your 2 squared down to - i² minus minus plus and 2? Plus i/6 is your first root.

[00:18:46]First okay? This is your first route coming up.

[00:18:53]Then the second root of this, if the second root is taken out, what will be the second? I told you that if your first root is coming in complex number. If it is in the form x + iy then its second root will be in the form x - iy.

[00:19:07]Ok? So the first root is your 2 + i 2 / 6 so what will be the second one? - i 2 / 6 is what we get.

[00:19:20]So you got two routes. Now it is asking you the value of A B C.

[00:19:23]What are you asking now? Asking the value of A B C. See what happens? Is there any problem? Can everyone hear the sound?

[00:19:33]Is the voice clear?

[00:19:36]Say.

[00:19:38]Ok? Let go. What to do now?

[00:19:41]What will we do first? You know two routes.

[00:19:43]Find the sum of both the roots. Let's assume. This route which is yours, we have accepted it as Alpha. And we accepted this, son. So find alpha + beta first.

[00:19:52]Alpha + Beta will be calculated first. So if you calculate alpha + beta then your -i is 2. Here +i is 2. Both will be cancelled out. 4/6 will come.

[00:20:02]Correct? Find the second alpha * beta. When we calculate alpha * beta, both will be multiplied. 6 * 6 in the denominator is 36, which we will multiply. 2 * 2 = 4 4 - i² * 2 How much will it take? 4 + 6 will come.

[00:20:19]6/36 Cancel this out 1/6 Now you know what a quadratic equation is? alpha + beta x + alpha * beta = 0 Here you will put the value 4/6 x + 1 / 6 6 will be your LCM. So this equation will become 6x² - 4x + 1 = 0 Whose value were you asking for? I was asking the value of a b c.

[00:20:48]Compare this to the quadratic equation. This is your a this is your sorry -4 is your b and one is your c. 6 - 4 and 1 option number one is correct.

[00:20:59]Is it clear? Move ahead.

[00:21:05]Ok. Let me tell you once again what has happened. I had given you a route. Your under route was -2. I changed this to a complex number.

[00:21:16]i² - 1 value kept. Solved this.

[00:21:19]Your first root came out to be 2 + i under 2/6, I assumed it to be alpha.

[00:21:25]What did you do for the second route? Whenever you have one root in a complex number, the other one will be its conjugate 2 - i 2 / 6 = beta then the value of alpha + beta is found to be 4 / 6 I can solve this from here also.

[00:21:38]2/3 comes. But here the same thing had to be done while keeping 2/3 and taking LCM. So I left it at that. Alpha * Beta 6 / 36 = 1 / 6. The value of put is compared with ax² + bx + c = 0 and we get the value of a b c.

[00:21:53]Ok?

[00:21:59]See next question.

[00:22:02]This question is about your assertion. You can also use the elimination method. If you can see the first statement clearly, you can eliminate it. The first and second equations, the second equation you have written has no real root. So first of all the question is telling you that one of the roots of the quadratic equation with real coefficient is 1/2 - 3i which of the following implications is true. Which one is true?

[00:22:30]So you can already see that if 1/2 is - 3i then the second root will also be a complex number.

[00:22:36]When you find its second root, it will also come in complex number.

[00:22:40]And if the imaginary roots of two complex numbers are coming then there will be no real root here. Will it happen or not? Will not done.

[00:22:47]Now option number second is correct. Now looking at the seconds in this, which one is it? It is in option C. Ok? Now we should know one more option.

[00:22:55]So what is the second root of the equation 1/2 + 3i? It will not happen because we will solve it now.

[00:23:02]We'll just write this root as 1/2 - 3i. We will multiply this up and down by 2 + 3i.

[00:23:10]If you solve 2 + 3i / 4 + 9 then this will come to you as 2 + 3i / 13 so where will you get the second root from? Will not done.

[00:23:23]First wrong second correct first is wrong first is wrong means your option has been eliminated your option has been eliminated and your second is correct so your second is only in this option but I will tell you the question how the question will be but you can see it by the method of elimination so you can solve it without lifting the pen you know first root is complex second will be complex but what is here it is given in the denominator, if you solve it it will never come second root second equation is clearly written no real root when both are imaginary root first is imaginary then second is also imaginary second one got eliminated both of these you know you applied option c but how will it happen I solved it 2 + 3i / 13 is coming what is this your this is first root alpha let us assume yours has come 2 + 3i /3 up 13 2 + 3i up 13 what will be your second root 2 - 3i up 13 what will you find out? That will become yours 2 - 3i /3 Now from here when you alpha we only have to verify one equation. So if we look at this verification, then there are only two of them. Your first route second route has arrived.

[00:24:36]So the real route is not being found. First second finish. Third, you have to check whether the equation is zero or not. Suppose you are not able to eliminate then at least solve it and do it. That's why I am telling you. So you take out alpha + beta from here. We will calculate alpha + beta.

[00:24:51]This 3i 3 - 3i will cancel out. It will become 2 4/13.

[00:24:54]We will calculate alpha * beta. Alpha * beta will become 2 * 2 = 4 4 + 9 13.

[00:25:01]13/13 * 13 is 13 times 13.

[00:25:07]Substitute this value into the quadratic equation. x² - 4/13 - 4/13x + 1 /13 = 0 Take the LCM of 13 here. 13x² - 4x + 1 = 0 means your third option is correct.

[00:25:26]Is it clear?

[00:25:28]Jai Hind. Jai Hind children.

[00:25:32]Ok? See next.

[00:25:36]You will get such questions. Many questions come up.

[00:25:38]But you have to do the same thing that you have been doing till now in the card.

[00:25:40]Correct?

[00:25:45]The quadratic equation 3x² minus k2 is not here. This is k². Not k2 This is k² k² + 5k x + 3k² - 5k = 0 has real root put say is real. Of equal magnitude is given to you b = 0. As soon as you see equal magnitude, it should come to your mind that your coefficient of x will be zero.

[00:26:11]You guys have asked so many questions.

[00:26:14]p = 0 you don't have to look at anything else. Where equal magnitude is seen, what will be yours and with opposite sign, what will be yours?

[00:26:22]p = 0 is saying opposite sign? What does opposite sign mean? Product Always Less Than Zero. Ok? In K Maths, things are related to each other from day one till a topic continues.

[00:26:35]Equal magnitude and opposite sign which one of the following is correct? b = 0 What was your equation here?

[00:26:45]What is the value of v here? This is the value of b. Put the value of b. -k² + 5x 5k = 0 From here take k common.

[00:26:55]Minus will become 0.

[00:26:58]5 + 5 will not be. Here in the equation I think -5. Let me check my question.

[00:27:29]Sorry, this wasn't your bracket. This is a bit misspelled.

[00:27:40]Here's the I think minus. There will be a minus here.

[00:27:47]-5 = 0 k = zero coming from here. And the value of k is coming out to be 5 here. Two values of k are found here. 0 5 What's Next? Opposite sign. Let's look at the product.

[00:27:58]What will the product be?

[00:28:01]What will the product be?

[00:28:03]Sun 0 is the opposite sign. Product less than 0 product what 3k² - 5k 3k² - 5k up 3 up 3 < 0 so if we solve it from here then 3k² - 5k equal sorry less < 0 less < 0 because you know what is there here less < 0 will become 3 0 or if we look at it from equality, from inequality then if your less than 0 is then one should be positive. Three is positive. There should be a negative. 3k² - 5k < 0. If you take k common from here then you will get 3k - 5 < 0 so here it is saying that k will be < 0 or k will be < 5 / 3. Ok? Sorry, it would be 3/5.

[00:29:09]This question is causing some problems.

[00:29:13]Ok? Let us look at one option.

[00:29:15]Rest I will check the question to see if it is a proper question. I think something is misprinted here. But I do n't have it right now. I will get it corrected.

[00:29:23]But let us look at it on the basis of the question at hand. So here k < 0 is coming. Here k = 0 is coming. So there is no such value given for k here. There is no such value given for k here. So no such value will exist which will verify the equation. Ok? We will leave this question here. The answer is correct, the process is correct. There is just some mistake in the question.

[00:29:45]I will take a look. I will look into it and get it fixed. Ok?

[00:29:53]This is your Army GD batch coming up. Ok? It is very important for the children who are preparing for Army GD.

[00:30:01]Only 99 is a limited offer. There are other things. We have faculty and teachers. Ok?

[00:30:11]Consider all the real roots of the equation x to the power 4 - 10x² + 9 = 0. What is the sum of the absolute values ​​of the roots? Here I am asking you what will be the sum of all the routes that come? Of absolute values.

[00:30:28]What does root mean? Like any of your roots came, three came, -4 came, -2 came.

[00:30:34]So taking its absolute value means that we will take its modulus. Will take the turn.

[00:30:39]What will happen if you take the turn? 3 + 4 + 2 = 9 but this is another example. not belong.

[00:30:49]This is your absolute concept. Let's see the question.

[00:30:52]What are you saying? Write down the question first.

[00:30:53]x² x 4 - 10x² + 9 = 0 What is the first thing to do? First of all we have to find its roots.

[00:31:01]So the quadratic equation is written. Here you put x² and y put x² = y then y² - 10y + 9 = 0 here y² - 9y - y + 9 = 0 here your y will go to common.

[00:31:23]Solve y - 9 - 1 y - 9 = 0.

[00:31:30]Your value of y will come out to be 9 + 1 root will come out to be 9 + 1 root will come out to be 1 Now these two roots have come out to be x² Put x² in place of y x² = 9 and x² = 1 From here the value of x will come out to be plus minus 3 And the value of x will come out to be plus minus 1 In the question I was asking you that sum of the absolute values ​​of the root In the question I was asking you to understand Big fan Hello Sir Hello Hello Hello What is the reply to the question you have asked? Do you have any questions?

[00:32:24]Let's continue the class. Ok?

[00:32:27]From here you got one or two values.

[00:32:29]Two values ​​came from here. What will happen now? If you want to find the sum then here it is 33 + -3 + 1 + - 1, solve it. 3 + 3 + 1 + 1 = 8 Your Answer.

[00:32:49]Ok?

[00:32:52]No question sir.

[00:32:58]Is this clear?

[00:33:01]Such roots were extracted. After finding the roots, by adding modulus to the absolute value, the minus sign will become plus and we will add it.

[00:33:07]Next if p and q are the non-zero roots. p and q are the non-zero roots of the equation x² + px + q = 0 then how many possible values ​​can q have? Asking you the value of q, how many possible values ​​can q have? And the equation is saying that p and q are also non-zero. Correct?

[00:33:29]What is an equation? x² + px + q So if you subtract p + q from here then you will get p + q - p this will give - p equation number one and if you subtract p * q from here then you will get p * q equation number second. Let's solve this.

[00:33:47]pq - q = 0 From here take q common.

[00:33:52]p - 1 = 0 tells you that q is non-zero. The question is telling you that q is non-zero. If q is non-zero then p - 1 will be 0. If p - 1 is 0 then what will be the value of p here? Forest. Take this value and put it into equation number one.

[00:34:13]Take the value of p and put it into equation number one.

[00:34:15]If placed in the forest, 1 + q = -1 forest will move there. Q = -2 Only possible value of Q is 1 -2 value is coming but what is the number of values? How Many is asking.

[00:34:31]-2 is not giving. Here the question is how many possible values? So how many possible values ​​only one. Is it clear?

[00:34:39]Move ahead.

[00:34:45]If 1 5x² + 26k + k = 0 the reciprocal of the other. Then what is the value of K? One route is a reciprocal of the other. One is alpha and the other is 1/alpha. The question ends there.

[00:35:01]Correct? What do you have to do? It is reciprocal.

[00:35:04]You need to find the value of K. So how will you solve this? You guys tell me the answer. There are people soon. Let me do it a little faster. Look, get the product out of it from here. I have given you two routes.

[00:35:22]Alpha * Alpha will be this product. So what will the product be here? k/5 k / 5 because c / a. k / 5 from here alpha alpha cancels out k = 5 is your answer. Option No. C Option No. C Next the roots of the B.

[00:35:42]Roots Alpha and Beta of the Quadratic Equation Satisfy the Relation Alpha + Beta = Alpha² + Beta² and Alpha Beta = Alpha² * Beta² What is the number of roots of the true quadratic equation.

[00:35:57]How many types of quadratic equations can you make with the help of these two equations or how many equations can you make? Correct? What do you do first?

[00:36:04]These are two equations. Ok? This is your equation alpha + beta = alpha² + beta square, leave it as it is, it has no role right now, come from this alpha beta = alpha² beta square, take it to the left hand side, alpha beta will come as 1 - alpha beta = 0, on solving, one value is coming from here, zero of alpha beta and the other value is coming from you, one of alpha beta, these two values ​​are coming from here, with the help of these two values ​​we will see how many equations will be formed in this. From here we will check once by keeping alpha * beta = 0.

[00:36:44]How many alpha + beta are coming. Here we will check by keeping alpha beta = 1. How many alpha + beta are coming. Let's solve this. Alpha Alpha + Beta equal to what is written? Alpha + Beta times squared -2 ​​Alpha Beta.

[00:37:04]This is written. Now this is 0 here. Put it.

[00:37:07]Put Alpha Beta = 0 will become 0.

[00:37:12]From here the value you will get is alpha + beta = whole square of alpha + beta, so bring it here.

[00:37:19]Alpha + Beta will go common.

[00:37:23]The value will come as 1 - alpha + beta = 0, so when you solve it, once you put it as 0 and once as zero, then after solving both of them you will get two values ​​of alpha + beta.

[00:37:36]Ok? From here you have to zero it once you have alpha. Alpha + Beta once it's zeroed. So from here you will get your two values ​​of alpha + beta.

[00:37:45]Meaning when your alpha * beta is 0 then you are getting two equations. How much are you getting? Two, from here, when we calculate alpha, alpha * beta = 1, let's move on to the next slide.

[00:38:03]This was your alpha + beta. It is given in the question. Now here when we put alpha * beta, we are moving at a little speed because two-three questions are left. Today I have to find the quadratic equation. Have to finish it. Ok? So whoever has any doubt can ask in the comment section.

[00:38:22]You will put this as one. What would you write to it first? Alpha + Beta = Alpha + Beta Squared - 2 Alpha * Keep it one. Alpha + Beta Squared - 2 1 Okay? Solve this.

[00:38:39]Alpha + Beta and what is there? -2 ok? So do this one thing at a time. Bring this here, bring this here. The left hand side of alpha + beta will not form a whole square or such quadratic equation.

[00:38:57]Let's take it to that side.

[00:39:04]is alpha + beta. Let's take it to that side.

[00:39:06]Alpha + Beta Squared - Alpha + Beta - 2 = 0 Now look here, this is the form of the quadratic equation written. If we replace alpha + beta with x here, what does it say? This is the equation you have written.

[00:39:25]= 0 This is what the equation is becoming. From here we will get two values ​​of x. You will get two values ​​of x.

[00:39:31]Means means alpha + beta, two values of alpha + beta will be obtained. So how many quadratic equations will be formed from here?

[00:39:38]Two equations will be formed from here also. Two equations were being formed by you and two equations were being formed by you. 2 + 2 will form a total of four equations.

[00:39:48]Ok? Is it clear?

[00:39:52]Alpha * Beta is 0 and Alpha * Beta is 1. The first equation involving both of them was kept in it and the other was satisfied. Two values ​​are available here. Alpha * beta two values from there get one here the total value will be four.

[00:40:17]But how many quadratic equations are there is the sum of the roots equal to the product of the roots. Just by looking at it, it is clear that it will be infinite. What value of alpha can you take? Now such equation is saying that how many quadratic equations are asking.

[00:40:29]This is equivalent to this. This equation is written by you. You can take any value of alpha here, alpha + beta here.

[00:40:37]will take the values ​​-1, -2, -3. A beta will exist for this. For this a beta value will be obtained. So there will be many such numbers.

[00:40:47]Where will you be able to count this?

[00:40:50]Suppose you are taking the value of alpha here as -1.

[00:40:52]So -1 + beta = - beta. If you take it out from here then you will get beta 2 beta = 1 beta = 1 / 2 Now here take the value of alpha -2 or -2 you will get beta something else, how much value will you get here finitely many, okay let's move ahead, yes please look at this question a little because I have not taught the log to you guys, it will come in the coming chapters, but you guys have already read it in the log, whenever we used to write the log in this form, then one thing was used here which I am writing.

[00:41:33]This formula has been used here.

[00:41:36]Because of this you should get the log. It was used. I have told this before.

[00:41:41]Look, the question is saying that this is a question asked in 2025.

[00:41:44]Very very easy.

[00:41:46]See what is the question? There were two questions here. Firstly, you have given this beta equation.

[00:41:56]Look, let alpha and beta be the roots of the quadratic equation x² + log 0.5 a². This is in base 0.5 x + log 0.5 to the base a² to the power 4 > = 0 where a² is not equal to 1 and log 0.5 to the base a² > 0. Asking you is saying that the value of beta² has been given to you. Saying what will happen to beta? So first of all, this is your quadratic equation.

[00:42:25]Your equation is x² in x and constants.

[00:42:29]So what will we do from here first? Let us first extract some values ​​from here.

[00:42:33]Which value will you extract? Alpha + Beta and Alpha * Beta are the solutions. What will happen?

[00:42:38]alpha * beta alpha + beta. So if you calculate the value of alpha + beta, what will be -b -b? - log 2.5 a2 a² up a so there is one here and if you calculate alpha * beta then what is the value of c/a c log 2.5 a² a² and its power what is its power? It is 4. Now what are you given? You are given ta², you are given beta square beta square alpha log a square 0.5, okay, first multiply this by beta, once the equation is sounding, sound is coming, everyone see, here is beta square, we will multiply this by beta, here also your alpha * beta will come and what will come to log a² 0.5? Here's your beta (what was the value of alpha * beta? log 0.5a² what was the base here? 0.5 what is the base here? a² so we can change its base.

[00:44:18]Either change its base or change its base. Let's leave it there in the power. Let's change its base.

[00:44:24]So we will put the value of alpha * beta here.

[00:44:27]This one which came log 0.5 a² power 4 into, we will change it. How will we change it? Before using the formula that I just told you, to do it one upon one, we will change the base with a number. 1 / log 2.5 2a² 2 a² Okay? So here both of these are the same. Here it is four power. One power will cancel out with the power. What will be left here?

[00:45:03]= log 2.5 a² power 33 power cancel equal to what will be left? log 5 to the base a² Okay?

[00:45:18]Many people get this question wrong because here But there's a bit of confusion about the base.

[00:45:24]Many people have entered the wrong options after getting the question right. Here, the base is a² and it's 0.5. All the options seem to be the same in the question.

[00:45:30]Okay? Keep this in mind. The value of beta has come up.

[00:45:35]Look at the same question in the next question. First, you asked us yesterday what to do for the NDA batch? This is your NDA second batch coming up in September.

[00:45:46]It costs only 299. The entire faculty is there. Good teachers will teach you. There are live classes. There are video classes. You'll get complete math and GATE preparation, live interaction classes, recorded lectures for revision, study notes, regular mock tests, doubt-solving sessions, SSB, and interview guidance. You'll get the full package.

[00:46:07]Okay? And we were given the contact number for this. You also have our app on the website. You can check it on the app. The last question I just looked at The second part of that was, what is the relation between alpha and beta?

[00:46:34]What is the relation between alpha and beta? So, what is the relation between alpha and beta?

[00:46:39]Look here, alpha + beta is alpha + beta.

[00:46:42]And what is written here? Whose value is this? This was the value of beta. Yes, look, the value of beta that came up was this. It was the value of beta. So, we can put the value of this beta here. So, alpha equals this. The value that came up was this plus this plus, right? We will put the value of beta here.

[00:47:22]What is the value of alpha plus beta here? Log 0.5a² plus log 0.5a² equals to minus log 0.5a², right? If we take this to the right-hand side and add it, it becomes -2 times. So, from here, the value of -2 times beta will come up. This is your relation. And what was in the question?

[00:47:51]-2 times alpha = -2 times beta. Right? That's the question.

[00:48:02]This is the last one. Now look, the question is asking, " What is the number of real solutions?" The mode has been given. But there's a slight change here. There's a slight change in the condition here, so the values ​​are given here.

[00:48:13]You haven't been told that there's no solution here. Till now, in all the questions with real solutions, you would have closed your eyes and answered, "No solution," okay, the equation doesn't exist.

[00:48:23]But here, there's a slight change. Here, it's asking you, "What will be the number of real solutions?" The inequality is given here. So, what is the modulus here? It's written as x² - x - 6 > = x + 2. You have to solve it and find the value. Once you find the positive sign, it means plus minus x² - x - 6 > x + 2.

[00:48:49]Okay? Once you take the positive sign. What will happen if you take the positive sign and solve it? It's a quadratic equation. You'll get two roots of order two.

[00:48:59]Once you find the negative sign, you'll get two roots of order two. What happens if we solve for the sign by taking the sign? You'll get two answers. But the problem here is that we ca n't add four here. We have to solve this. The common rule also has to be found. We wo n't add the common twice, right?

[00:49:16]So, the common goods will also be found here. So, let's solve this and see what happens.

[00:49:22]Time is over. But this is the last question. So, let's see.

[00:49:28]Here, we'll take the positive one first. x² - x - 6 = x + 2. This is given by x². Then - 2x will come here -8 = 0 x² - 4x - + 2x - 8 = 0 x will go common x - 4 + 2 x - 4 = 0 Then x = 1 will come 4 one will come -2. Now, we'll solve by taking the minus - x² - x - 6 = x + 2. This gives -x² + x + 6 - x - 2 = 0. Here, x - x² and the minus plus cancels out.

[00:50:11]Your + 4 = 0 value will come here, so x² = 4 x = + - 2. The value comes from here. Two comes from here, but -2 is common to both, so if we count it once, the value of x will be -2 2 4. A total of three values ​​are coming.

[00:50:30]Okay? The medium is also Hindi medium.

[00:50:36]Both Hindi and English medium, and I will try to ensure that Hindi is spoken fluently in your classes starting Monday, because, son, I'm a little comfortable with English in math, but I will try to teach in Hindi. That's why the questions on your slides are in both Hindi and English.

[00:50:54]Okay? And this was your last class on quadratic equations. You should do as many questions as possible, as many practice sets as PYQs, as many concepts as possible, everything has been explained. Now your only job is to thoroughly read the PPT I taught, prepare all the questions concept-wise, and practice thoroughly. Take mock tests.

[00:51:18]This is most important, you should prepare at least the NDA exam before the exam.

[00:51:24]After the first preparation, you should give 15 to 20 full length mock tests.

[00:51:28]Okay? Give the mock test. In the NDA second batch that we are bringing, you will be given a proper mock test. There is also SSB interview.

[00:51:37]Your lecture class is also for revision. There are study notes, practice material. Regular mock test is very important.

[00:51:46]And we are giving you all this faculty for Rs. 99.

[00:51:52]Okay?

[00:51:55]Doubt Sir. Yes. You have a doubt, tell me what is your doubt?

[00:52:00]Private, private doubt, what kind of doubt is it? Please hurry up, I have my next class, what is your doubt? Okay, let's stop the class here today and we will meet on Monday with a new topic. Okay? Jai Hind.

[00:52:33]Yes, question or sir, I studied NDA 2026 from you.

[00:52:46]See you in the next class. From the next class.