00:00 Hello everyone, my name is Farahini E.C and I am your Mathematist teacher for today.

00:22 The topic before us is Choretic Equation and we have the objectives here.

00:31 In this case, you need to be able to identify the sum and product of roots of Choretic Equation.

00:39 At the same time, you need to be able to form a Choretic Equation if you are giving the roots.

00:46 Having said that, we have examples we would like to treat here.

00:53 Look at the question one here.

00:56 If you have one of the roots alpha to be -1/2, excuse me, one of the roots is alpha and is -1/2.

01:14 This question is saying that you will find the Choretic Equation kind of giving 8x^-bx-3.

01:24 Wow, 8x^2-bx-3 is equal to zero.

01:32 This b is unknown.

01:34 So how do you form Choretic Equation giving all this?

01:37 How do you find the roots?

01:38 When the Choretication has an equation that is not known.

01:42 But it is very simple.

01:44 From the principle we had the other time, the sum and product of a Choretic Equation.

01:51 That tells you that if you have the general form of Choretic Equation to be a x^2+bx+c=0, remember

02:01 quite well that the sum really sum alpha+bx is given as -b/a and the product as discussed

02:23 is alpha/bx=c/a.

02:31 With this understanding you can go on step by step and get out your b given a kind of information

02:41 of what your alpha is.

02:44 And here we go, we can now look at 8x^2-bx-3=0.

03:03 In this case you divide through by your 8 so that you can have 8x^2/8-bx/8-3x/8=0.

03:21 When it is done you can now start to see how this comes out.

03:30 It is caught off here so this gives me my sum and this gives you my product and I can easily

03:41 fix it.

03:42 I have for you alpha+bx that is sum is equivalent to alpha+bx=-b/a.

04:00 So, don't forget I need to recall quickly the sum is actually -bx+bx=b/c.

04:19 In this case your c is 8 and you have -b/a.

04:23 So have you said that you can now proceed to see what you can make do with this.

04:31 If alpha+bx=8 you can make this one equation 1, alpha+bx=-b/8 you can call it equation 1.

04:45 So that fascinating how to get out of it and your product is this as I've said.

04:53 So you can now write product alpha better=-3/8 you can make this one equation 2.

05:08 It becomes clearer.

05:09 I remember very well that the value of alpha is actually -1/2 okay.

05:14 If that is taken you can go ahead and substitute quite well for the value of alpha and for

05:20 me I can go on substituting for the value of alpha I have -1/2 times alpha.

05:28 I mean times beta is equal to -3/8 and the minus we can still out there.

05:35 So I can now have my beta to be well if I write properly so you can see what I have here

05:46 is actually better over 2=3/8 and what you do here in order to get beta is you multiply

05:57 with that by 2.

05:58 If I do that I have beta over 2 times 2=3/8 times 2.

06:07 It's clear now that your beta will be what? 3/4 so the second root is now 3/4.

06:23 Now that you have it you can now go ahead to get your B. Do not forget that your sum is

06:32 actually -B/8 which is your equation 1.

06:37 Go ahead and substitute for the value of beta in equation 1 and that brings out your B

06:44 quite well and this is what we do.

06:46 If you have that alpha + beta is equal to -B/8 you have come up with what you have.

06:58 As alpha is -1/2 excuse me alpha is -1/2 all right and beta you just got it to be 3/4.

07:18 So with that you can go ahead and write properly now -B/8.

07:28 So for me if you put your LSM together you can have 4 RSEM, LSM is 4 there.

07:40 2 in 4 - 2 times -1/2 -2 + 4 in 4 there's 1 times 3 that gives you 3 there.

07:52 So this go ahead to put B in a better perspective for you.

07:58 So you can go ahead and sum up your fraction here to have -2 + 3 I give you 1/4 equals -B/8

08:12 and with that you can come up with something decent.

08:17 So if you have this that can lead you to B is equal to you know what I'm going to do?

08:28 I multiply by 5/8 so let me do a clear job for you.

08:35 If I multiply by 5/8 okay that can lead me to 1/4 times 8 okay.

08:45 I mean -B/8 times 8 so it costs away I can have -B to be equal to 2 and eventually if I

08:58 want to get my B I cannot write B excuse me I cannot write B equals -2.

09:07 Yes so if you have -B equals 2 do not forget you have to find

09:14 B so that gives us a B equals -2.

09:21 So a very quick one again is that we shall now go ahead and find what's the value of this

09:34 expression will be okay.

09:37 Look at this expression it says if you find find 1/r4 -1/b everything squared all right

09:54 a good way to do this is that you have already got your alpha to be -1/2 and you have your

10:06 B to be 3/4 so if that is done just go ahead and substitute into it however you need to

10:16 come up with some ideas I think from here the sum will have to come out and the product

10:24 will have to come out and then you get it done.

10:27 So for me finding the else some of this I have I mean find the LCM what do I have?

10:36 I have alpha better okay and in this case I am going to have lcm that gives me beta plus

10:48 alpha which is even as alpha plus beta everything squared okay over alpha beta okay everything

11:02 is squared all right so for me what I have here is the outcome of the expression of

11:12 quadratic equation with sum as alpha beta I'm alpha plus beta I'm product as alpha

11:20 beta so if I do that I have alpha plus beta squared over alpha beta squared I do not

11:32 have problem with this because I know that the sum from what we had before now if you

11:38 remember very well I have the sum to be -B/A and since my B is -2 and my A is 8 I can

11:53 go ahead to know what my sum what my sum will be and what my product will be.

11:59 So with this I have alpha plus beta squared this gives me alpha squared plus 2 alpha beta

12:14 plus beta squared all over alpha beta squared yes it's becoming big and big but you can

12:26 always bring it down now that we have the value for our sum alpha plus beta and we have

12:36 a value for our product alpha beta do not forget that we have also gone for that to find

12:42 the value for B to be -2 we can go ahead to find the value of any roots we are given

12:49 to find for instance in the example that we want to try saying find the value for

13:01 find the value for 1 over alpha minus 1 over beta everything squared so you can go ahead

13:16 this is squared so you can write the first bracket 1 over alpha minus 1 over beta 1 over alpha

13:29 minus 1 over beta so where you get that all you do is expand and expanding is quite easy

13:38 you know the first term here goes to multiply this and this goes to multiply this in the same

13:43 way you multiply this by this and in the same way to this go for this you put everything

13:50 together then things happen let's go so if I do the first expansion 1 over alpha times

13:56 another 1 over alpha that gives me 1 over alpha squared and I can follow on few to have

14:03 minus 1 over alpha beta and for this I can come over here to have a minus 1 over alpha

14:14 beta as well and finally minus times minus gives me plus so I'm going to have plus 1 over beta

14:22 squared I can do a good thing to rearrange so I can come up with a good thing and that is

14:29 pull this first and the last together and the middle one together so when I do that I can have

14:35 1 over alpha squared plus 1 over beta squared and I have if I pull this together what do I have

14:45 the LCM remains alpha beta but I have 1 over this 1 over this I'm going to have 2 of it so I have

14:55 minus 2 over that minus 2 over alpha beta and when I get that done a good way to look at it now

15:07 is how do I get the expansion and move on how do I get the expansion and move on and that

15:16 takes me to the real expansion of returns but I still have a thing to do here if I find here

15:23 or some of this that gives me alpha plus beta everything squared alpha plus beta everything squared

15:34 I mean alpha beta everything squared then I can go on I have beta squared minus alpha squared

15:45 I mean plus rather and this one remains I do not need to bring it together with it

15:53 yes and gradually I come up and write an expression that we lead me out of it

16:01 this gives me alpha squared plus beta squared over alpha plus beta squared minus

16:10 I can always take care of this because I know the value of my product to be 3 over 8

16:16 but I can still relax and fix it later on so in this case I have something I can deal with here

16:27 and that gives me the sum of this this leads me to what I have as alpha plus beta squared minus

16:43 2 alpha beta the expansion of this is this okay and if I have that I can already see what comes

16:52 out of it for me if I fix it this one in here look at what I'm going to have

16:59 or shout instead of writing this alpha squared plus beta squared I'm going to write all these and

17:07 they fix the product on it so with that is it don't deal and what do I mean I have alpha plus

17:16 beta everything squared minus minus 2 alpha beta all over alpha beta squared don't forget

17:29 minus 2 all over alpha beta so with this I can go ahead and fix my values appropriately

17:38 by the side I know that my sum is b over 8 and my b is minus 2 so I can have one over 4 squared

17:57 okay minus I know my what it is alpha beta which is my product to be 3 over 8 all right so I can

18:10 fix in here and write 2 into 3 over 8 it's becoming clearer I remember very well that my product

18:19 is 3 over 8 so I write 3 over 8 squared and I can go ahead to write minus I remember very well

18:31 that my alpha beta is actually 3 over 8 so I have minus 2 over 3 over 8 so with this in view I can

18:42 go ahead and find the result at the value of my 1 minus alpha 1 minus I mean 1 over alpha minus

18:52 1 over beta so here I go this one gives me 1 over 16 minus 2 cuts 8 here I have minus 3 over 4

19:02 and all over 9 over 64 for my 3 over 8 squared and minus 2 over 3 over 8 what I do here is what

19:19 we refer to as reciprocal you invert it okay by inversion I can have minus 2 times 8 over 3

19:30 all right so this now leads me to something fantastic that I can do it right away yeah now that we have

19:40 subset for this sum and the product of our roots we can go ahead to simplify the fractions

19:48 and come up with a value for 1 over alpha minus 1 over beta all squared and this is this is it

19:56 LCM of 16 and 4 that gives me 16 over I mean 16 LCM 16 here 1 times 1 that's 1 and 4 in 16 that's

20:08 4 times 3 that gives me minus 12 and I know for the fraction is

20:16 been done by inverting when you have fraction to have fraction what you do is you invert by much

20:25 implication and that gives me 64 over 9 minus this leads me with minus 16 over 3 so with this

20:36 something comes up and I have 16 cuts 64 here that gives me 4 and for 1 minus 12 I have minus 11

20:46 excuse me minus 11 over 9 because I eventually have 1 here so the format applies 11 later

21:00 and I'm going to have that so minus 16 over 3 yes so getting that evaluated you can now see that

21:14 you're going to have a minus 44 over 9 minus 16 over 3 so with this your LCM remains 9

21:26 and you can go ahead to write minus 4 4 minus 16 which eventually will lead you to getting

21:34 minus 60 over 9 as your answer so the value for 1 over alpha minus 1 over beta everything squared

21:43 is actually minus 60 over 9 and the last one is very simple we have the equation given as

21:51 6 x squared and the minus sorry 6 x squared

22:02 minus k x plus 48 equal to 0 if yes this is very simple you remember your sum remains minus

22:17 k over 6 and your product remains 48 over 6 but the interesting thing about this question

22:29 is that enough information is giving you because one of the roots is the square of the other one

22:35 so just make a pre-mind which of the roots are you going to use alpha if alpha is the first root

22:40 then the second root is a collineant beta from the image information given the question you say

22:47 is alpha squared so for me some becomes alpha plus alpha squared then for me product becomes

23:00 alpha times alpha squared with his alpha cube so very easy in the equation now you can now

23:09 see how you can translate the from here you have into value of getting the k yes I mean into the

23:17 way of getting the k so if you have your product to be s alpha cube alpha cube is equal to

23:29 48 over 6 and for you know you can get that to be 8 and if alpha cube is going to 8 so therefore

23:42 your alpha is actually 2 because 2 is the cube of 8 so I mean the cube root of 8 is 2 or 2 is the

23:55 cube of 8 so if alpha is 2 so the second root we now be what 4 because it's a square of it so

24:04 we can go ahead and now say therefore the second root is 4 being the square of the first one

24:18 if that is giving you to find k becomes easier because you can write the sum is equal to

24:29 alpha plus alpha squared is equal to 4 I'm sorry is equal to 2 plus 4 is equal to 6

24:48 don't forget you have your sum to be equal to 6 now if your sum is actually k over 6

24:57 sum is k over 6 so you now come up to say if sum which is k over 6 is also equal to 6 then

25:12 what is going to be the value of k therefore k over 6 is equal to 6 and therefore k is equal to 36

25:23 so with this you have come to realize that k is 36 when the roots are 2 and 4 so with this you have

25:36 been taught how to get the sum and product of a quadratic equation if you are given the equation

25:42 or how to get the quadratic equation if you are giving the roots so I will now leave you with

25:46 evaluation question that we let you know how far you have gotten the explanation that is 1

25:55 find the sum and product of the roots of 8 x squared plus 4 x minus 24 is equal to 0 also 7 x squared

26:02 minus 28 x plus 12 is equal to 0 and also number two if you are given an equation to x squared plus

26:08 x minus 12 is equal to 0 you need to find the value of 1 over alpha plus 1 over everything squared

26:16 thank you for watching my name is Farahimi EC till next time