POLYNOMIALS [Theory of quadratic equations of alpha & beta] #maths #mathstricks #university #jamb

追加: 2026-05-12

[00:00:00]Okay, in today's class we are going to continue from part two of our previous video on theory of quadratic equation.

[00:00:11]Okay, but basically we are going to be talking about questions from alpha beta.

[00:00:18]So if they ask you a question like this if alpha and beta are the roots of the polomial 2 x 2x^2 - 5x + 2 = 0 what is the easiest way you are going to solve this problem for those of you that is writing CBT remember you have to share and subscribe be to our YouTube channel for amazing videos like this. Okay, the first thing we have to do is to identify the coefficient of the quadratic equation.

[00:01:00]The coefficient of this quadratic equation is having that a is 2, b is -5 and c is 2.

[00:01:21]Once you know the coefficient of this quadratic equation, you are done with the solution. So you bring out your calculator by making the work easier for you. You press mode 5 3.

[00:01:37]Mode 53 can assist you in solving this quadratic equation to get the value of alpha and the value of beta. So the best thing you do is press mode 53. You press 2 equal to the value for a. You press minus 5 equal to the coefficient of b. And you press two. You press equal to the value for C. Finally, you press the final equal to to get the value of your alpha to be two. Then you press another equal to to get the value of your beta to be 1 / 2. Listen, when you solve quadratic equation using your calculator, it helps you to know the root of this equation provided they say is alpha and beta. That simply means the first root is alpha, the second root is beta. So if you have two and 1 / two as the solution to the problem that simply means that 2 is alpha which is the first value and 1 / 2 is beta which is the second value. So what do they tell us to simplify? They say you should simplify alpha all over beta plus beta all over alpha. Since we know the value of alpha, alpha all over beta plus beta all over alpha. Since we know the value of alpha which is two, we substitute the value have 2 all over the value of beta is 1 / 2 plus beta is 1 / 2 and alpha is 2.

[00:03:27]Moving on, when we press this thing in our calculator, you discover that the whole thing here is four plus the whole thing here is 1 / 4.

[00:03:39]Okay. So, which will end up giving us our final answer. This * 16 + 17 / 4. We check our option if there is something like that in the option is option number D. Okay. The easiest way to approach any question on alpha and beta is by getting the value of alpha and getting the value of beta. So after getting the value you go to the question ask and substitute your parameters and get the answer. Okay. In question two, [laughter] in question two they said if alpha and beta are root of x^ 2 - 3x + 1 = 0.

[00:04:31]Then alpha minus beta is what?

[00:04:39]Okay. Using the same approach, you press mode 53.

[00:04:48]Then the coefficient of a here is one.

[00:04:52]You press one. You press equal to. The coefficient of b is -3.

[00:04:58]You press minus3. You press equal to.

[00:05:02]The coefficient of the constant is one.

[00:05:05]You press one. You press equal to. Then finally you press the final equal to to get alpha.

[00:05:14]You press the final equal again to get better 3 +<unk> 5 / 2 I is I there.

[00:05:34]>> Okay. Beta is what?

[00:05:39]3 -<unk> 5 / 2 >> like this.

[00:05:48]>> Okay.

[00:05:54]Okay. So, what did they tell us to find?

[00:05:56]They say you should find alpha minus beta. So, punch all in our calculator. Alpha - beta will be giving us 3 +<unk> 5 / 2 - 3 -<unk> 5 / 2 then you press to press this thing in equal to you just have to keep it in the same way it is in this place when we collect common denominator we have to we have 3 +<unk> 5 minus we multiply the whole thing here - * 3 is - 3 - * +<unk> 5.

[00:06:39]So this will go with this. Root 5 +<unk> 5 is 2<unk> 5 all over 2. Finally 2 will go with 2. Our answer will become roo<unk> 5 which is option number.

[00:06:58]Okay, that is for question number two.

[00:07:01]But listing and always the most important thing is for you to always use your calculator and get the two roots which is alpha and beta. When you get the two roots you narrow your interest to what they are asking the question they asking becomes very simple because you have already gotten the value of alpha and the value of beta. Okay, that is all for example one and two. So moving on, let's talk about question number three.

[00:07:31]In question number three, they said the root of 2x^2 - 7x + 4 are alpha beta. Find the equation whose roots are alpha all over beta comma beta all over alpha. Okay, the sweetest way to do this thing is to write the equation down. 2x^2 - 7 x + 4 = 0. Okay, you get the value of alpha and get the value of beta. How are you going to do it? With the same step, you identify the coefficient of a, b and c by having a to be = 2, b to be = -7 and c to be = 4. So the first thing I'll do is to press mode 53.

[00:08:32]I'll press two equal to I'll press -7 equal to I'll press 4= to then I'll press the final equal to to get the value of alpha which is 7 + <unk>7 / 4 and the value of beta is equal to again to get the value of beta to be 7 -<unk> 17 all over 4 like this right okay so for us to evaluate remember in our previous video I taught you how to find a quadratic equation whose root is given. How to do it? If you did not watch the previous video, you should do well to go and watch it. We use the formula of finding equation sum of the roots * x plus product of the roots equal to zero. So this thing is telling us to sum the roots and to multiply the roots. So using your calculator you will get the sum of the roots. Sum of the roots to be adding these two roots together. 7 + sol 17 all over 4 + 7 - 17 all over 4. Then you get product of the root in similar way with your calculator by having 7 + 17 all over 4 * 7 + - 17 all over 4.

[00:10:35]Okay. To to evaluate the sum of the roots and the product of the roots, you have to plug this thing in your calculator. 7 + roo<unk> 17 all over 4 plus adding the two roots together. So you observe that when you add these two roots you will get 7 / 2 and when you multiply the two roots you will get two.

[00:11:01]Okay. So when you plug it in the original equation that we are looking for, you see that the sum of the roots we got it to be 7 / 2 * x plus product of the roots gave us 2 when we multiplied it all equal to zero. So to get the quadratic equation since this one is fraction we have to multiply everything by the denominator which is 2. So multiplying x^2 with 2 we have 2 x^2 multiply - 7 all over 2 by 2 the denominator will go it will be leaving - 7x plus multiplying 2 by 2 here I will have 4 = multiplying 0 by 2 I will get zero.

[00:11:56]So this is the equation that the root is uh alpha all / 2 beta all / 2. So you check your option. Is there any option that looks like that? Yes, option number B is correct. Okay, that is all for number three.

[00:12:17]Number three, you just have to get the values of the roots. You use this main equation that connects sum of the root and product of the root. Get the sum of the root and product of the root. Then you find the equation. That is all. For number four, simply you will do the same thing that we have been doing before.

[00:12:43]For number four, you do the same thing.

[00:12:49][laughter] We said if alpha and beta are the roots of the polomials.

[00:12:59]Okay. Obtain the value of 1 all over alpha + 1 all over beta. Okay. Let me use [laughter] a simple a common way to teach you this.

[00:13:11]For those of you that is writing y and those of you that are writing neo and watching this video from afar, we have uh x² + x + 1 = 0. So in the study of alpha and beta there is what we call sum of alpha beta which is given by the formula minus b all over a And we have what we call product of alpha beta that is given to us as c all / a. Okay, this one we don't want to press alpha and beta with calculator.

[00:13:53]Let's just abandon that for now. So here we have a here to be one. B here to be 1 and c is still one. So minus b all over A is going to be -1 all over 1 which is -1.

[00:14:13]Then C all over A is going to be 1 all over A is what? One which will still give us what? 1. So we have the sum to be minus one and we have the product to be one. So moving on, they said we should find 1 all over alpha plus 1 all over beta. Finding the LCM. LCM is the denominator which is alpha beta.

[00:14:39]So alpha divided by alpha beta will remain beta * 1 will give us beta plus beta into beta alpha will remain alpha * 1 will give us alpha. So you observe that the numerator is the sum and the denominator is the product and we have the sum to be minus1 all over we have the product to be one -1 / 1 is -1 which is function number a so most of times if you asked this type of question the only thing you need provided that the root is alpha beta you have to get the sum and the product then simply simplify it.

[00:15:22]That is all for example four.

[00:15:25]So for example five they said if alpha and beta are the root of the polomials this obtain the value of alpha squ + beta squ. Okay let's go over to our calculator method approach to see how we address this question. The equation given to us is is 2 x^2 - 5x + 7 = to zero. So remember that the coefficient of a here is 2.

[00:15:59]Coefficient of b here is - 5. The coefficient of c here is seven. So the good thing that we are going to do is to to use our calculator. Press mode 5 3.

[00:16:21]You press the coefficient of a2. You press equal to. You press minus 5. You press equal to. [laughter] You press 7. You press equal to. You press the final equal to to get alpha and you press another equal to to get beta 5 / 4 + i. Then the value of beta is what?

[00:17:03]5 5 / 4 -<unk> 31 / 4 I. Okay. So the question said alpha squ + beta squ. But for you to get alpha squar and beta square. Since is a complex root, you see this equation have complex roots. You must put your calculator in complex number mode by pressing mode two.

[00:17:34]When you press mode two, you enter Q in exactly what you have here. You are queuing it in as 5 / 4 +<unk> 31 all over 4 I all 2 Plus plus you see we have alpha squar here plus beta squared we'll be having 5 / 4 -<unk> 31 all over 4 I all squared remember they are telling you to square alpha + square betas since you have gotten the value of alpha to this you square it plus you have gotten the value of data to be this you square it finally you get the final answer remember what we use in pressing I this imaginary I in our calculator is ng in our casual calculator original casual calculator so this one fraction we use this this symbol in calculator to press fraction so when you press fraction actually press 5 all over 4 + root. Root have this symbol in your calculator.

[00:18:56]So you have to locate this sign to press root. To press I, you press E.

[00:19:02]Okay. Uh once you are done entering it, you see that the answer will give you - 3 over 4. Okay. Which is option number C.

[00:19:17]Okay. In number six, they said if alpha and beta are the roots of the equation x^2 + x - 6 x 2 + x - 6 = 0. You know that a here is 1, b here is 1 and c here is -6.

[00:19:45]Okay. So the first thing is to get the value of uh alpha and the value of beta by pressing mode 53.

[00:19:54]Press mode 53.

[00:19:59]You press one = you press 1 = press - 6 = then you press the final equal to to get the value of your alpha which is two and you press the another equal to to get the value of beta which is -3.

[00:20:25]Okay, what is the question? I say get alpha minus beta all squ. So the good thing to do is to have alpha minus beta all is alpha which is 2 - beta which is - 3 all 2.

[00:20:44]- * minus is + 2 + 3 is 5 all squared will give us 25 which is option number D.

[00:20:55]Okay, moving on.

[00:20:58]Number seven, they said if alpha and beta are the root of the quadratic equation 4 x^2 - 4x + 1 = 0.

[00:21:12]They said then alpha cube plus beta cube is dash.

[00:21:18]The first thing is you press a is 1, b is -4 and c is 1. So you get the value of alpha and the value of beta.

[00:21:32]Okay. With your calculator mode 53 you get alpha. Alpha is >> [laughter] >> <unk>3 2 + 2 2 - <unk>3 2 + <unk>3 and 2 -<unk>3. Okay, for us to get alpha cube plus beta cube, we have alpha cq + beta cube becomes 2 + <unk>3 all cub + 2 - <unk>3 all cub which you punch it exactly like this in your calculator to get 52 which is option number A. Okay. So the trick about this particular alpha and beta equation is knowing the value of alpha and knowing the value of beta.

[00:22:40]Sometimes the value of alpha and beta goes to imaginary roots or complex number roots which you have to put your calculator in complex number mode mode two to figure it out. So when you get the value of alpha and beta when the two roots of the equation then you narrow your mindset on the question asked that will help you to evaluate the answer.

[00:23:04]Thank you for watching this video.

[00:23:07]Uh we are going to do factors and remainders theorem in our next video so that we finish theory of quadratic equation and now study more and deep on the concept of other concept lines.

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