00:02 Good day, my name is Adedokun Afeez.

00:16 I'll be your further maths teacher,

00:17 and the topic is root of quadratic equation.

00:20 At the end of the topics,

00:26 students will be able to know the sum and product

00:29 of the root of quadratic equations,

00:31 form quadratic equations given sum and the product of roots,

00:35 nature of the roots, which are equal roots,

00:38 re-root and no roots.

00:40 Then our identified conditions for a given line

00:44 to intercept a curve, be tagine to curve,

00:48 not intercept a curve.

00:50 Now let's take a look to sum and product of root

00:55 of a quadratic equations.

00:59 In SS1, we thought what is a quadratic equation is.

01:04 So if you have an equation of the form,

01:08 a x squared plus b x plus c equals zero,

01:13 we know that this is what we call quadratic equation.

01:17 And the value of x, when x has a particular value,

01:22 and that is what we call root of quadratic equation.

01:25 So if, for example, we are able to solve this quadratic

01:29 equation, and we have the x to be of the form x equals

01:34 minus b plus square root of b squared minus a c all over

01:44 2a, all x equals minus b minus square root of b squared

01:50 minus 4a c all over 2a.

01:53 So this is the root of quadratic equation.

01:57 But how do we, the sum of the sum and the product

02:02 of the root of quadratic equation?

02:03 Meaning that if you have these two roots together,

02:06 and we also multiply these two roots together,

02:09 then we can have the sum, and we can have the product.

02:21 Now let's assume, let's, let's how far equals minus b

02:30 plus square root of b squared minus 4a c all over 2a.

02:36 And let's beta equals minus b minus square root of b

02:42 squared minus 4a c all over 2a.

02:45 We can find the sum of these root of equation.

02:49 We can find the sum of these root of equation

02:54 when we have these two, that is, alpha plus beta.

02:59 Meaning that it's going to be minus b plus square root

03:03 of b squared minus 4a c all over 2a plus minus b

03:10 plus minus b minus square root of b squared minus 4a c

03:17 all over 2a.

03:18 So we find the sum of these.

03:20 So we have 2a, the sum is 2a.

03:23 So we have minus b plus square root of b squared minus 4a c

03:30 plus minus b, the plus we change to minus.

03:33 So we have minus a.

03:35 So this is going to be minus b minus square root of b squared

03:41 minus 4a c all over 2a.

03:43 So when we, you see that this is going to be,

03:47 we can't do this, just we can't do this.

03:49 So what we have left is minus b minus b.

03:51 And that's what gives us minus 2b all over 2a.

03:56 And so we can't do 2 also.

03:57 We have minus b all over a.

04:00 So it means that the sum of the root of quadratic equation

04:05 equals minus b all over a.

04:08 Therefore, alpha plus beta will give us minus b all over a.

04:16 So it means whenever you are given a quadratic equation

04:18 of the form as squared plus bx plus c equals 0.

04:22 And you have to find the sum of the root and the product

04:25 of the root.

04:25 So we have to use this for the sum.

04:28 So let's take a look to the product as well.

04:29 So when you have, it means we are going

04:33 to multiply this alpha beta.

04:35 So it means minus b plus square root of b squared minus 4a

04:40 c all over 2e times minus b plus as minus square root

04:48 of b squared minus 4a c all over 2e.

04:52 So when you multiply this together, we are going to have,

04:55 we're going to have minus, we're going to have plus c all over a.

05:01 By the time we multiply this together, we give us c all over a.

05:05 So it means, therefore, we have alpha plus beta, which

05:09 is the sum of the products.

05:12 I mean, which is the sum of quadratic equation?

05:14 We have it to be minus b all over a.

05:17 And we have the product to be c all over a.

05:21 Now, let's take a look to this example.

05:33 We have example number one.

05:35 Find the sum and product of the root of the quadratic equation,

05:39 3x squared minus 5x plus 2 equals 0.

05:44 We are given a quadratic equation, which

05:46 is a 3x squared minus 5x plus 2 equals 0.

05:52 You can see that equals 0.

05:54 So we are going to recall the general form of quadratic equation,

06:01 which is a x squared minus plus b, x plus c equals 0.

06:07 You can see that if you compare these two quadratic equations,

06:10 we're going to have our a to be what?

06:12 Our a is 3, and our b is minus 5, and our c is 2.

06:18 So it means, and we have to find the sum and the products.

06:21 So the sum of the root is going to be alpha plus beta.

06:26 And that is minus b all over a.

06:29 And our b here is minus 5.

06:31 So we have minus minus 5, all over a is 2.

06:35 So minus times minus, we give us plus.

06:38 So we have 5 all over 2.

06:40 So this is the sum of quadratic equation.

06:44 So the product to, so alpha beta, we have c all over a.

06:48 Our c here is 2, and our a here is, our a is 3.

06:55 So we have 2 all over 3, and that is 2 over 3.

06:59 So it means the sum and the product of this equation

07:02 give us 5 over 3 and 2 all over 3.

07:06 Now, we can also find quadratic equations

07:18 when the sum and when the roots of the quadratic equation

07:24 are given.

07:24 So if, for example, you are given the roots

07:27 of quadratic equation alpha, alpha plus beta, alpha,

07:32 let's alpha and beta be the roots of quadratic equation

07:44 of quadratic equation.

07:46 We can also find the quadratic equation

07:49 of these by given the root and the sum.

07:54 And the formula to use is x squared minus the sum alpha

07:59 plus beta x plus the product alpha beta equals 0.

08:04 So that is how to find.

08:05 So if, for example, in this example and about 2,

08:08 if you are given, find the quadratic equation.

08:11 Who's root are, or if you go 1?

08:14 We have minus 1 and it's minus 1 and it's.

08:20 So it means our alpha a is minus 1 and our beta a is 8.

08:26 So if you want to find the sum yet, we have alpha plus beta.

08:30 So it means this is going to be 8 minus 1.

08:36 And that will give us 7.

08:38 And if you want to find the product, we have--

08:41 that is going to be 8 times minus 1.

08:45 And that will give us minus 8.

08:47 So in that case, we have to find the equation of this root.

08:57 So it means therefore it's going to be x squared minus alpha

09:02 plus beta x plus alpha beta equals 0.

09:07 And that will give us x squared minus our sum.

09:11 It's give us 7, x, and our product give us minus 8 equals 0.

09:17 So it means therefore we have x squared minus 7,

09:20 x minus 8 equals 0.

09:25 So it means when you are given, it roots the root of equation.

09:30 And you have to find the quadratic equation.

09:33 So we can also find the quadratic equation

09:35 using the root has weight.

09:40 Now let's go to 2 romantic or 2.

09:45 We have our alpha here, alpha is 1 over 2.

09:54 And we have beta to be 5.

09:56 So we can find the sum of this.

09:59 And that the sum is going to give us--

10:01 we have 1 over 2 plus 5.

10:06 1 over 2 plus 5 gives us 11 over 2.

10:10 And alpha beta, we have 1 over 2 times 5.

10:15 And that will give us 5 over 2.

10:16 So we can easily find the quadratic equation, which

10:19 is x squared minus alpha plus beta x plus alpha beta

10:24 equals 0.

10:26 And we know our sum to be 1 over 2.

10:29 So we have x squared minus the sum is 11 over 2.

10:37 11 over 12, I mean 11 over 2, x plus 5 over 2.

10:42 So we multiply through by 2.

10:44 We have multiplied through by 2.

10:46 We have 2 x squared minus 11 x plus 5 equals 0.

10:52 So this is the quadratic equation.

10:55 Whenever you are given the two roots.

11:05 Now, I'm not saying that we want to talk about

11:08 is the nature of the roots.

11:11 When you have the roots, it's very, very important for us

11:14 to know the nature of the root also.

11:17 Whether the roots is a number, whether it is not a number,

11:20 whether it is a thousand number.

11:22 So it is very, very important for us

11:24 to also understand the nature of the roots.

11:27 We have whenever you have a quadratic equation.

11:29 Now, let's recall the roots.

11:33 Let's recall the roots that we have x

11:38 equals minus b plus or minus square root of b squared

11:46 minus 4ac, all over 2.

11:50 And let's d equals b squared minus 4ac.

11:57 What determine the nature of the roots

12:00 is b squared minus 4ac.

12:03 That's what actually determine the roots.

12:06 I mean, the nature of the roots.

12:08 So that means we have x equals minus b plus or minus square root

12:14 of b square root of d, square root of d, all over 2a.

12:21 Now, we need to note these three conditions

12:25 for the nature of the roots or for the value of d.

12:29 Because what we say, what determine the nature of the roots

12:31 is what b squared minus 4ac.

12:35 Now, whenever d equals b squared minus 4ac.

12:39 So it means the first one is when your d,

12:42 the value of your d is greater than 0.

12:45 What type of the roots are we going to have?

12:48 What type of root are we going to have?

12:50 If your d greater than 0, then the root of d quadratic equation

12:56 is going to be real and distinct.

12:58 Whenever our d is greater than 0, we're

13:02 going to have the real and distinct roots.

13:06 And the graph of this, this graph explain what it means by real

13:11 and what it means by our distinct roots.

13:15 If you look at this graph, this is the graph of quadratic equation.

13:19 We have a greater than 0.

13:21 It means that when you are given--

13:24 if, for example, we have x squared, a x squared plus b

13:30 x plus c equals 0, your a must be greater than 0 for you

13:35 to have a distinct root, your a must be greater than 0.

13:42 As you can see in the graph, our a must be greater than 0.

13:45 It means our a can be 1, it can be 2, it can be 3, it can be 4,

13:49 it can be any number.

13:52 So that is when you have-- then your d also, if your a greater than 0,

13:56 your d must also be what?

13:57 Must be greater than 0.

13:59 And this is where this-- we have this point.

14:04 It is where the value of our root of equation

14:05 can also be calculated.

14:07 So we have our x equals alpha 1.

14:10 And that is for the first root.

14:13 In the correct equation, you must have two values.

14:16 So and the second one, x is beta 1.

14:19 So it means this is the second value of what?

14:22 Second value of your x.

14:25 Then also, if your a is less than 0, then your d must also

14:29 be less than 0.

14:30 And we also have the roots of the equation for that as well.

14:34 So when you have d greater than 0, the type of--

14:41 the nature of the roots is going to be

14:43 real and distinct roots of quadratic equation.

14:49 Then also, when d is less than 0.

14:53 The second condition is when your d, when d is less than 0.

14:59 When d is less than 0, you see that we're going to have in--

15:05 we're going to have-- if d a is less than 0,

15:07 we're going to have a negative number, a negative number.

15:10 And that means the square root of negative number

15:12 is going to give us a kind of imaginary number.

15:15 And that is an advance or mathematics anyway.

15:18 But what we are trying to say is that what type of or what

15:21 nature of the roots of the equation are we going to have?

15:27 So the roots of these, when d is less than 0,

15:32 then the roots are not real.

15:35 The roots are not real.

15:36 They are not real.

15:36 They are not number.

15:37 They are not distinct.

15:38 They are what.

15:39 They are said to be imaginary because the square root of d

15:44 is not a real number because we have a negative number.

15:47 So it is not going to be what?

15:48 Because when you-- if, for example, our d,

15:51 like we said, our d is less than 0.

15:53 And we have d to be minus 5, for example.

15:56 Square root of d is going to be square root of what?

15:59 Minus 5.

16:00 The square root of minus 5 will not give us a real number.

16:03 It will give us a real number.

16:05 So it means it's going to give us an imaginary number.

16:08 So when d is less than 0, always note that the d,

16:13 nature of the root is going to be an imaginary number.

16:18 Now, we can show this in the graph.

16:21 You can see that the straight line will not be

16:24 to intersect where the curve.

16:27 Where the curve-- where we have the curve.

16:31 And you can see that when you're going to have,

16:35 it will not touch the graph as well.

16:41 So it means our a here-- the graph of this form

16:45 is a is greater than 0.

16:46 Like we said, in our a will be greater than 0.

16:49 Then if h greater than 0, d is going to be less than 0.

16:52 Then if we have d less than 0, our a is going to be less than 0

16:56 as well.

16:56 So when d is less than 0, the type of ordinate of the root

17:03 we're going to have is going to be an imaginary.

17:07 So also we have-- the last one is what we have.

17:14 When d equals 0.

17:15 So when d equals 0, we have the real and equal.

17:18 And they are said to be co-incidenta.

17:21 So the type of root, or the nature of root we're going to have,

17:24 is going to be a co-incidenta nature of the roots.

17:28 So we also show the graph of that.

17:30 So this is a graph, a is greater than 0, d equals 0, a is less than 0,

17:35 and d also equals 0.

17:37 So since d enable us to find the position of the graph,

17:41 why relates to x-asix?

17:43 It is called a discriminant.

17:45 So it means that our d is called discriminant.

17:50 So we could show the example of what we are--

17:54 what we just said now, the nature of the roots.

17:56 And that is, the first one is we have x squared minus 3x

18:02 minus 2 equals 0.

18:03 And you can see that we're going to have d.

18:06 Our d is going to be b squared minus 4ac.

18:09 Don't forget we said that d determines the nature of the roots.

18:12 So our a a is 1, b a is minus 3, and c is minus 2.

18:19 So our d is going to be b minus 3 squared minus 4 times a.

18:25 Our a is 1 times c minus 2.

18:28 So we're going to have 9 plus 8, and that is 17.

18:35 So you can see that d is greater than 0 now.

18:37 So the type of roots we're going to have here

18:39 is going to be real and distinct.

18:41 Then the second one is we have x squared minus 6x plus 9

18:53 equals 0, and we have a a is going to be 1, b is going

18:57 to be minus 6, and c is going to be 9.

19:00 So if we have our a, our d equals b squared minus 4ac.

19:09 And we can see a is minus 6 squared minus 4 times we have a.

19:15 Our a is 1, then c is 9.

19:18 This is going to be 36 minus 36.

19:21 So we have 0.

19:23 So you can see that d equals 0 now.

19:26 So what type of root are we going to have?

19:28 We're going to have a real and equal root when we have d equals 0.

19:37 So the example of battery, let's say we have 2x squared

19:55 minus 2x plus 5 equals 0.

19:58 You can see that our a is going to be 2, b is going to be minus 2,

20:03 and c is going to be 5.

20:04 When compared with the genera form of quadratic equation,

20:11 and that is-- and then our d is going to be b squared minus 4ac.

20:17 And our b is minus 2 squared minus 4 times 2 times 5.

20:23 And that is 4 minus 8.

20:26 This is 4c.

20:27 So we are going to have minus 36.

20:29 You can see that our d is less than 0, a, d is less than 0.

20:34 So and what type of root are we going to have?

20:36 What nature of root are we going to have when our d is less than 0?

20:40 We're going to have our imaginary roots.

20:43 So these are ways in which we can easily find the nature of the roots.

20:50 And we can also note that we can find the type of the root,

20:55 whether the sum or whether the product.

20:58 So far, we have talked about the sum and the product of the roots.

21:03 And we have shown how we can find the quadratic equation whenever

21:09 we are given the roots of the quadratic equation.

21:11 Then the last one we talked about is the nature of the roots.

21:15 The nature of the roots tells us that when--

21:19 what the term is nature of the root is d.

21:22 And that d is what we call discriminants.

21:24 And d equals b squared minus 4ac.

21:28 And when d is greater than 0, what type of root are we going to have?

21:32 We're going to have real and distinct whenever d is greater than 0.

21:36 When d equals 0, we're going to have real and equal roots.

21:41 Then when our d is less than 0, I mean, it's less than 0,

21:47 we're going to have an imaginary roots.

21:51 So this is the way in which we can find--

21:54 we can determine the roots of quadratic equation

21:57 as our topic implies that it is the root of quadratic equation.

22:00 So we only want to talk about the roots of quadratic equation

22:05 because we have talked about the quadratic equation in SS1 class.

22:10 And this is what describes what determines

22:13 the words even explain the root of equation.

22:19 And the evolution has been given.

22:22 In all words, we have talked about so far.

22:24 The first one is if other than beta and the root of the equation,

22:28 we have 2 x squared minus 2 x plus 5 because 0,

22:32 we have to find the sum, a sum of the roots.

22:36 And we have to find the alpha beta.

22:39 That is the product of the root.

22:40 And we have to find our alpha plus 1 over beta.

22:48 The second one is find the quadratic equation.

22:50 Those roots are 3 over 5 and 1 over 2.

22:54 That is, we can find quadratic equation

22:57 whenever we are given the roots.

22:59 So we can easily find that also.

23:01 So the last evolution is determine the nature of the roots

23:05 of the equation s squared minus 5 x plus 2 equals 0.

23:11 Thank you for watching.

23:12 My name is Adidu Kama-Fiz once again.

23:14 Till next time.