00:02 Hi, everyone.

00:14 My name is Fawahimi EC,

00:18 and I'm your Math Teacher for this session.

00:22 The topic we're gonna have this time around

00:29 is finding the roots of a quadratic equation,

00:34 and also forming a quadratic equation given the roots.

00:40 The objective of this lesson is that

00:45 it should be able to identify the sum and product

00:50 of a quadratic equation.

00:53 At the same time,

00:54 it should be able to also form a quadratic equation

00:59 if I'm giving the roots.

01:02 Having said that, we like to have some introduction.

01:08 If I have alpha as one of the roots,

01:16 and the second one, better, all right?

01:21 What that means is I say the root X is equal to the value alpha.

01:26 And I can also write the root X is equal to the value beta.

01:35 So I have two roots.

01:42 And in our process, I can write X minus alpha equal zero.

01:51 I can also write X minus beta equal zero.

01:56 Recall that for roots of a quadratic equation,

02:02 you can either have one of the factors being equal to zero

02:09 or both are zero.

02:12 So for me, I can write X minus alpha

02:19 into X minus beta.

02:23 And I have equal to zero.

02:27 When I have this, I can expand.

02:32 And when I just done, what happened is,

02:37 you have to follow the process we had before.

02:40 The X here can go to multiply the X here,

02:43 and the X here can go to multiply this.

02:48 In the same thing, the alpha here we go to multiply this.

02:52 And lastly, the alpha we go to multiply this beta here.

02:58 So there are various ways you can get to expand this,

03:03 but this about the easiest way.

03:07 So now let's see what happens when we have this.

03:15 I have X squared for my X times X.

03:20 And for my X times minus beta, I'm gonna have X minus X beta.

03:26 If I go to expand using the minus alpha to multiply,

03:34 that is minus alpha times X, that gives me alpha minus alpha X.

03:40 And if I go to multiply minus alpha and minus beta,

03:44 I'm gonna have plus alpha beta equal zero.

03:49 Things happen here as saying I can group this

03:55 to in order to factorize my X.

04:00 And when I just done, I'm gonna have X squared.

04:03 What's up?

04:06 I have minus X beta, I have minus alpha X.

04:12 So what that means is that I'm gonna have minus sign outside

04:17 and I have alpha plus beta then X outside.

04:22 A good thing can, is it me figure out what I have here?

04:27 Because if you expand back, you can have minus alpha X.

04:32 You can have minus beta X as the minus sign goes into multiply.

04:42 And with that, you can have alpha beta equal to zero.

04:47 So this now leads you to what we have.

04:52 We can write out exactly what the sum and product

05:00 of quadratic equation will be.

05:09 In my mind, if I recall very well, I remember that

05:14 the equation of X is one, okay?

05:17 Then the equation of X gives me minus

05:22 bigger part in minus alpha plus beta.

05:30 And this goes for my sum.

05:39 Okay, and the last one, alpha beta, that goes for my product.

05:46 So what we do here is that given the two roots alpha beta,

05:54 you have to establish that the sum could be given us

06:00 minus alpha plus beta and the product could be

06:05 gotten to be alpha beta.

06:08 Still in continuation of this, you recall quite well

06:13 that the general form of a quadratic equation is actually

06:17 A X squared plus B X plus C equals zero.

06:22 And in the light of that, you can easily compare coefficients.

06:35 If I want to, maybe I should just bring you into this.

06:40 Look at, if I have this to be a quadratic equation,

06:45 as I just put on the board for you.

06:49 And I have said this is also a quadratic equation.

06:53 X squared minus alpha plus beta plus X plus alpha beta.

07:03 What comes to your mind is that the question of X here

07:08 being one is equal to A.

07:13 And the question of X here being B is equal to minus

07:18 a bracket to be alpha plus beta.

07:20 And the constant term C here is equivalent to alpha beta.

07:25 So with this, you can always come up to,

07:31 now write properly that this expression

07:36 that you've gotten X squared minus alpha plus beta,

07:44 X plus alpha beta.

07:50 If you see that your X squared is having one as its coefficient,

07:59 then all you need to do is you have one as the coefficient

08:04 of your X here, if you divide through by A,

08:08 then you will left with X squared plus B over A X plus C over A.

08:13 With this, you can convince anyone that, yes,

08:25 the sum is equal to B over A is equal to minus alpha plus beta.

08:30 And make a pardon, alpha plus beta,

08:44 and your product will be C over A X squared,

08:55 over A equals your alpha beta.

09:00 It's very simple when you have a question to apply it with,

09:08 so that you can only stand very well.

09:13 We're gonna try some of this example to actually drive it home

09:17 and know that you can always form a quadratic equation

09:21 if you have the roots.

09:23 And vice versa, if you have a quadratic equation,

09:28 you may be asked for the sum and product of the roots as well.

09:33 Now, looking at the first question we have yet,

09:36 it says that you should form the quadratic equation

09:41 whose roots are five and four.

09:43 If you're gonna get that done, you can look at it this way.

09:51 You have your roots to be five and four.

09:56 That is, let X be equal to five and X equals four.

10:03 Okay, so for me, here I can have, therefore,

10:21 I can have X minus five equals zero.

10:26 And in the same way, I can have,

10:32 okay, beg a pardon, I think what I have is minus four, okay?

10:37 So that we don't run into some problems.

10:39 We wanna have X equals five and X equals minus four.

10:44 All right, we can also,

10:49 we can also have it before anyway,

10:52 but this is what we have.

10:53 If I have that, in this, in the same vein as well,

10:59 I'm gonna have X plus four equals zero.

11:03 So these are the two factors for this equation.

11:08 I did not know that I want to form.

11:09 I want to form.

11:10 So, I now use the principle that either,

11:16 the two of them are zero or one.

11:19 So, if the two are zeroes,

11:22 that means I'm gonna have X minus five

11:26 into X plus four equals zero.

11:31 So when I have that, what comes out is this.

11:35 In the same vein, I can expand.

11:39 You can just write for your teacher to know, expand.

11:44 So your teacher will know that, okay,

11:46 you really want to expand.

11:48 So if that goes for that, and this goes for this,

11:51 and your minus five goes to or multiply that,

11:56 and minus five goes to multiply that,

11:58 they move out and get it done.

12:01 X times S gives you X squared.

12:04 And X times four gives you plus four X.

12:10 Minus five times X gives you minus five X,

12:15 and minus five times four gives you minus 20,

12:20 equal to zero.

12:23 You can collect like times, all right?

12:26 And I give you X squared, excuse me.

12:30 I give you X squared minus X,

12:38 as in four X, minus five X, gives you minus X,

12:43 minus 20.

12:44 So that leads to the quadratic equation

12:49 that has it's roots as five and minus four.

12:52 Very easy.

12:53 The other one is like it as well.

12:57 It says find the equation to whose roots are

13:06 two of a five and minus one number one over two.

13:10 So if you're gonna do that,

13:12 it's the same way you figure out the fact that yes,

13:17 we can also say let's X be two of a five,

13:23 and also let X be minus one over two,

13:31 one number one over two.

13:35 If you have that, what comes to your mind first?

13:38 It's two, fours to four,

13:40 equal to zero by transferring to the left side.

13:46 So for this one, I have X minus two over five,

13:51 okay, and for this one, I have X equals bigger pardon,

13:57 X plus one, one number one over two.

14:06 Equal to zero, equal to zero.

14:09 One more thing I can do before I bring them together

14:13 is to express this one as proper fraction.

14:15 That I can do by writing X plus three over two,

14:21 I've seen two times one plus two, that gives you three.

14:25 So if you have that, you can now go ahead

14:33 and put it together, that is, you have X minus two over five,

14:38 two over five, okay,

14:50 into X plus three over two.

15:02 All right.

15:03 Well, with this, you can as well expand and come out neatly,

15:10 that gives you X squared for X times X,

15:15 and for this other one, X times three over two,

15:18 you can just make it three X over two,

15:23 and you now go for minus two of five times X,

15:28 that gives you minus two X over five.

15:32 And the last one minus two of five times three over two,

15:38 that gives you minus, okay, six over 10.

15:43 Well, some other guys can do the division right away,

15:47 I mean, off hand and write it,

15:49 but then nothing stops if you still write it as it is.

15:53 That's up six over 10, six over 10, equal zero.

15:58 So if you'll have this, all you need to do now

16:01 is collect like terms and come up with something.

16:04 I have three X over two, I have five,

16:10 I mean, I have two X over five.

16:14 So for me, I have X squared plus,

16:19 the other term of my two and five gives me 10,

16:23 so I can have two in 10, five,

16:28 five times three, that gives me 15 X, minus.

16:33 And if I have that, I cannot have,

16:37 well, I cannot have,

16:41 well, I begin to look at this,

16:46 I think we have something here.

16:51 Our roots are actually minus five,

16:57 minus two over five, and minus one over one,

17:02 oh, number one over two.

17:07 So if I have this to be minus then,

17:12 I should have, excuse me,

17:14 I should have gotten this to be plus, all right?

17:18 So that we have it in the same way.

17:22 Obviously, if the two are negative negative,

17:26 then you're gonna have positive factors.

17:29 And in that line,

17:31 well, the thing is that we can always correct that,

17:34 and I guess we're plus here, I guess we're plus here.

17:37 So you work out the factors such that you ensure

17:42 you have the right sign, okay?

17:47 And what I've done here is to correct the fact

17:50 that our roots are minus two over five,

17:54 minus one over one over two, okay?

17:58 So we can now go on to see the expansion

18:02 of the roots you've been given,

18:05 and see how it leads you to a quadratic equation.

18:09 Yes, so here, I'm gonna have a plus 10,

18:14 as in five in 10, I mean, I'm gonna have four other,

18:18 as in five in 10 is two times two x, actually four x.

18:23 So, yeah, I was making a point at a time.

18:26 If I have six over 10, I can now divide through by two.

18:30 So if I have two in six x, me three,

18:34 and two in 10 x, me five.

18:35 So I can have three over five equals zero.

18:40 So with this, I come up with my equation properly,

18:45 and I can have x squared plus 15 x plus four x,

18:51 that leads me to 19 x over 10.

18:55 And if I have plus three over five equal to zero,

19:00 then I can now begin to come up with my stove,

19:06 and that leads me to a multiplication of 10

19:11 through the terms, so that I can clear the fractions, all right?

19:18 So if I multiply through multiply true by 10, okay?

19:23 That leads me with 10 x squared, okay?

19:35 Then if I have to multiply 19 x over 10,

19:45 then a 10 goes away, that's exactly what we want to achieve here.

19:48 That gives me 19 x.

19:51 If I multiply three over five by 10,

19:55 then that leads me with three times two,

19:58 and that's what we want to do, we'll look clearly fractions.

20:00 So you'll have six.

20:01 All right, so with this, equal to zero,

20:07 you have been able to form a quadratic equation

20:11 given the two roots that are minus over five,

20:16 and minus three or two.

20:18 So the other example, you can also try.

20:22 Well, let's look at this or the one,

20:26 and then we can talk about it, and then see what we can do.

20:30 Look, the question too is talking about

20:32 given the quadratic equation, that's the other way around now.

20:38 Given the quadratic equation,

20:40 you're asked to find the sum and product of the roots, right?

20:46 Well, for the first one,

20:49 I think I'm going to leave you with that,

20:51 but let me try the second one,

20:53 so that you can know how to get out of this.

20:56 So the second one is talking about six x squared,

21:01 equal to two x two x plus 20.

21:16 Yeah, if this is what you have,

21:20 and you like to get the sum and product of this,

21:27 then all you need to do is rewrite.

21:45 If you rewrite this, that gives you six x squared,

21:50 as in the two x now comes to the left hand side

21:53 to give you minus two x,

21:56 equal to the 20 comes to the left hand side

21:59 to give you minus 20 equals zero.

22:02 All right, a good way to look at this is,

22:07 you divide through by two, okay?

22:14 And see what, I mean, you divide through by six

22:19 and see what comes up.

22:20 And the reason what you do is that you want to ensure

22:23 that the coefficient of your x squared is one.

22:28 And that is what leads you to the principle of sum and product.

22:35 Don't forget, the sum and product of quadratic equation

22:38 is given as x squared minus r for plus beta,

22:43 bracket plus x, then plus x, I mean, plus alpha beta.

22:47 So the coefficient of your x squared must always be one.

22:52 You know, to achieve that, we divide through by six, right?

22:57 And if you're doing that, that is a divide through

23:02 by six.

23:07 When I get that down, that gives me x squared,

23:11 minus two x over six, minus 20 over six equals zero.

23:16 Well, the next thing is try to see how to simplify your fractions

23:24 so that you don't deal with like numbers.

23:26 So that gives me x squared, two over six could lead me

23:32 to minus one over three.

23:34 So I have x over three, all right?

23:37 And 20 over six could lead me to 10 over three,

23:43 and that would be okay for me.

23:45 As in, simplify the numbers so that they become simpler.

23:50 All right, if you have this,

23:52 then you can now go to compare coefficients of your quadratic terms.

23:57 I have x squared minus x over three,

24:05 minus 10 over three equals zero.

24:08 What does this mean in terms of some of the product of factors

24:13 that gives me x squared minus alpha plus beta x

24:18 plus alpha beta equals zero.

24:29 When I have this, quickly I can now see what to do with this.

24:35 That means my sum is equal to minus alpha plus beta equals

24:45 to minus one over three, all right?

24:48 And then that means alpha plus beta is equal to one over three.

24:55 And lastly, the product is alpha beta, okay?

25:05 Is equal to minus 10 over three.

25:08 So you can try this one with the subsequent one.

25:14 And in order to one of this session,

25:20 I'll leave it with this information.

25:22 Questions one, form a quadratic equation,

25:24 whose roots are minus five and three,

25:28 four and minus one, two.

25:30 Find the sum and product of the roots of eight x squared plus four x minus 24 equal to zero.

25:36 And seven x squared minus 20 x plus 12 equal to zero.

25:40 Thank you for watching.

25:43 My name is Feime EC, till next time.