Introducing the roots of the quadratic equation
as a topic this is kissing academy
The roots of a quadratic equation you may
call them the solution to a quadratic equation
My first of all the quadratic equation is
arees in the form ax squared plus, BX plus
C
The value of a B and C are
Constants now if we are to find the solution
to that quadratic equation we shall use that
formula
The X is equal to negative B plus or minus
under the square root b squared minus 4ac
over 2a
Now in that bulldozer method the method that
we use to solve a quadratic equation
the term b squared minus 4ac
is referred to as the discriminant and
the discriminant is what determines the nature
of the roots of the quadratic equation so
Now let's explore the possibilities here,
so if B
Squared minus 4ac this part of the formula
if it is equal to zero it means that the roots
of that quadratic equation are real and
are repeated if b squared minus 4ac is equal
to zero it means that the roots are real and
repeated and
therefore important negative B over
2a
Now if b squared minus 4ac is greater than
zero then the roots are different
it means that either X is negative b plus
or minus b squared minus 4ac over 2a or
X is negative B. Minus the difference lies
here. One is a positive one is a negative
right there
Now if B. Squared minus 4ac is less than zero
then the roots are imaginary
Why are they imaginary because now if this
part of the?
Formula is less than zero it means we are
going to find the square root of a negative
number
And since the square root of a negative number
does not exist
It only means that the answer we shall get
there the root of that equation will just
be
Imaginary in essence if b squared minus 4ac
is less than zero then
The roots are imaginary since there is no
quantity whose square root is negative, so
let's explore some examples here
Some illustrations. He has said whether the
following equations have
two real and distinct roots
Real roots or repeated roots lie so you got
room on one?
here
Room on one is full x squared plus 3x minus
12
Is equal to 0 we know that 4 v squared?
- 4 SC which is the discriminant should be
equal to?
We now find out. This is a according to the
general formula. This is a this is B. This
is C
So we substitute in here. What is B squared
B. Squared is 3 squared minus?
4 times a which is 4 times
C which is negative 12, so
We shall end up with
Now of course you are going to get a negative
answer and nine minus that negative answer
We are going to end up with a positive answer
on the positive as a way to get here is 201
So since b squared minus 4ac is 200 1 it means
that B squared minus 4, AC is greater than
0 and
according to our previous
Assertions here we say that if B squared minus
4 is is greater than 0 then the roots are
different
It means that the roots are they are 2 and
they're different it means that in this case
for this particular number
We shall conclude by saying
that B squared
minus 4 AC is greater than 0 therefore the
roots
You can say that there are two real roots
that are distinct
let's go through mine -
Ramon - is 4x squared plus
8x plus 4 is equal to 0 we want to find the
nature of those
Like before we say this is our value of a
that's our value of B
That's our value of C and now with us. We
seek to find with the nature of the roots
So it's still going to be B squared minus
4ac
This is going to be B squared. That is 8 squared
minus 4 times a which is 4 times
C which is 4 so that's going to be 64
That is what it means it means that the equation
has repeated roots as simple as that
If eight x squared minus 8 X plus P is equal
to 0 has a
Repeated root to find the values of P. So
if you have to find the values of P
We know that for repeated roots. Just like
we solved before
Discriminant which is b squared minus 4ac?
is
Going to be a four to zero
For repeated roots so for a number like this
for us to find the value of P
If that equation has repeated roots it means
you're supposed to simply
substitute in the expression
Using that equation so that we have what to
get the value of P sees that we have been
told the third equation
Has repeated root, so we know that eight is
the value of a that is negative eight is the
value of B and?
positive P is the value of C so B squared
is negative 8 squared
Minus four times a which is eight
Times C. Which is P?
That should give us zero
It's good for a different kind of numbers
We are going to look at numbers that are going
to require us to find the range of values
of
Certain variables within quadratic equations
this example is asking us to find the values
of lambda
for which the equation x squared minus 6x
minus 1 plus lambda into 2x plus 1 is equal
to 0 may have real roots
Let's rewrite that equation first x squared
We want to find the range of lambda now if
you find a range of lambda which
Will be such that this equation has real roots
if it has real roots
it means that the discriminant which is b
squared minus 4ac
Is greater than or equal to zero so it means
you're supposed to rearrange this equation
after rearranging that equation we substitute
the terms a B and C in the expression B
Squared minus 4 AC is greater than or equal
to 0 we say greater than or equal to 0 because
that is the condition
For this equation to have real good after
substituting that there
Lambda two x plus lambda
Is going to be equal to zero x squared?
We collect like terms
We put the expressions with X alone that is
minus 6x
plus lambda 2 X
Then we have plus
lambda minus 1 is going to give us 0
so that is giving us 0
So we'll end up with the quadratic expression
in X is going to be x squared minus put this
minus X
Into 6
plus lambda minus y
Is equal to 0 so we formed a quadratic equation
in X by quadratic equation in X. I mean, this
is our value
This is our value of B
This is a value of C
We are looking at a quadratic equation in
this form that
this quadratic equation has got real roots
means that the B squared minus 4 ay
C is greater than
Or equal to 0 so what is B squared in this
case B?
squared is 6 minus 2 lambda, so we shall go
ahead and say I
in bracket 6 minus 2 lambda close bracket
squared subtract that from 4
Times the value of a the value of a in this
case is 1
Actually 2 the coefficient of x squared which
is 1?
times C the value of C is lambda minus 1 times
lambda minus 1
Is greater than or equal to 0?
So if you see with our calculation we shall
end up with 6 minus 2 lambda
into 6 minus 2 lambda
Minus this is going to be 4 into lambda minus
1 is greater than or equal to 0
- 28 lambda
Plus 4 lambda squared is going to give us
better than or equal to zero if we divide
through by before
We shall end up with 10
So I'm going to end up with a quadratic equation
here in lambda
It's either a the lambda is equal to 2 or
Lambda is equal to 5 as far as our question
is concerned the find values of lambda for
which this equation
May have real roots so we are supposed to
test for which range
Will satisfy that equation to have real roots
So we have here that the value of lambda is
2 less than lambda less than
5 yeah, we have here the value of lambda is
greater than 5
Then the value of lambda here is less than
2
So let's take a number below - let's pick
0 and
If we pick 0 and we substitute it into that
expression that supports this expression
Of course this expression has been simplified
to this
This is the expression the discriminant that
says that b squared minus 4ac is greater than
0 it has been simplified to this
so if we just substitute in there and we see
if
this range of that range or that range will
support this a
session
So if we pick a figure there are two, they'd
still pick 0 0 minus 2 here
will be negative 2 negative 2 times negative
5 if this is 0 of course you got your times
negative 5 will give us a
Positive 10, so he asked ourselves is positive
10 greater than or equal to 0 yes
So what does that mean it means that X less
than 2 it satisfies that range?
The common test here 2 3 4 5 let's just pick
3 here
if we come on substitute 3 in this expression
3 minus 2 3 minus 2 is 1
So I'm here 3 - 5
3 minus 5 is giving us negative 2
So this is 1 times a negative 2. It's going
to give us a negative answer here a
Negative answer is definitely less than 0.
It's not greater than 0 so it means this range
from 2 to 5
Does not qualify so we proceed to lambda is
greater than 5, let's pick 6
if you come and put 66 minus 2 we have 4 so
and here 6 minus 5 is 1
So 4 times 1 is 4 is 4 greater than equal
to 0 yes, so it means this region satisfies
that
that inequality
So what does that mean it means that now you
conclude by saying that there are your values?
You
I mean your values here are simply lambda
is less than two comma and also lambda is
greater than negative five
Find the range of values for M for which that
is rule so we have our equation there
It is x squared plus, MX
Plus M is equal to zero, but you know that
for real roots
That applies for your route so in this case
our value of B is going to be M squared
minus 4 times a which is 1
Times C. Which is M is greater than or equal
to?
0 so you continue with arithmetic
M squared minus
4 m is going to be brutal than or equal to
0
Now for work. What are therefore we start
looking for the range of values that will
satisfy
The real roots, so draw a simple small line.
We have 0
And we have 4
so this is the region where m is greater than
or equal to 4, this is the region where m
is?
Less than 0, and this is the region where
we have 0 being less than
Or equal to M less than or equal to 4 so now
which region is going to satisfy this inequality
So the range of values of M
Let's pick negative 1 here if we put negative
1 negative 1 times negative 4 is negative
5 negative 5 times negative 1 is positive
5 positive 5 is it greater than or equal to
0 yes
So it means that this range of values is going
to be supported this
Supports the inequality so we proceed to this
range between 0 to 4, which is your 4 this
one here
Let's choose 1 1 minus 4 is negative 3 negative
3
times 1 is going to give us negative 3 still
is negative 3 greater than or equal to 0
No, it's not so since it is less than it means.
It's not going to support so that range does
not support
We go to this M is greater than or equal to
4. Let's choose 5
5 minus 4 is 1
1 times 5 is 5 is 5 greater than equal to
0 yes
So it means that this range is also going
to?
support that inequality
so if so after finding the range of values
that satisfy that inequality
We shall conclude by saying that
Buddies of em way that equation has real roots
our
M is less than equal to 0
and
M is greater than equal to 4. Those are the
range of values
Magic expression a quadratic equation is always
in the form of ax squared plus
BX plus C so for us to know whether a next
a quadratic expression has got a minimum point
for a maximum point
Will simply look at the coefficient of x squared
For a coefficient of x squared that is a positive
we know that that quadratic equation has a
minimum point break up with a minimum value
the coefficient of x squared is a positive
and
Wake up with a maximum value the coefficient
of x squared is or is a negative black which
I already see
Find the minimum value of 2x squared minus
6x plus 3
hence determine the value of x actually at
which it occurs
First of all it is a minimum value because
the division of x squared is a positive so
it's a minimum value
it has a minimum point so determine the that
minimum point we are supposed to complete
squares and
formulate that equation so that it is in this
form
Completing squares and
formulating this equation in form of a 2x
+ P in brackets squared
It is from them that we shall be able to find
the minimum value
the minimum value which will be the Q will
be so
Only when the terms in the brackets that is
x + B are equal to 0 like we shall see in
our subsequent
example for example here, 2x squared minus
6x plus 3 is equal to 0
So we pull the 2 out the brackets when we
pull the 2 out of the brackets
it's going to become 2 into x squared minus
3x plus 3 over 2 is equal to 0
So when we complete squares of this?
It means that we are going to get the coefficient
of x get the coefficient of x
Divide it by 2 after dividing it by 2
square the answer you get when you square
the answer you get you're going to add it
first to this term
at the answer you bitch to that term and then
subtract it from that value see I
Think was that so in this case we the coefficient
of x is going to be
3 so we divide 3 by 2, and we divide 3 by
2 will be get 3/2
3/2
Square the answer is going to become 9 over
4 so we go and add
9 over 4 to that term so it becomes 9 over
2 PI over 4 plus that the rest of that term
plus
Then we subtract that 9 over 4 from the value
of C
Which is 3 over 2 so it's going to become
3 over 2 minus 9 over 4 that's how we complete
the square
so
If you solve this quadratic equation we find
that this is a perfect square
So it's going to become X minus three over
two in brackets squared plus
6 minus 9 over 4
So our hands are here. We shall end up with
the expression 2
Into X minus 3 over 2 all in all in brackets
square
Minus 3 over 2 this is equal to 0 points.
We'll call it the function of X
So now the minimum possible value
Of the expression is going to become it shall
be negative 3 over 2 and
It will occur
When
X minus 3 over 2 is equal to 0 therefore the
value of x for which that expression is
Minimum is going to be equal to 3 over 2 so
We put in our answer the minimum value. Well
that quadratic equation is negative 3 over
2 and
It happens when the value of X is 3 over 2
cause this whole thing is supposed to be 0
the UH next time passes that find the minimum
value of the function
2x squared minus 4x plus 5 and step the values
of X for which it occurs. We are going to
use a similar approach
to what we've been doing before they are going
to complete squares and
Make that equation
The question is 2x squared minus 4x plus
5
Pull the 2 out of the brackets. It's going
to become 2 into x squared minus 2x plus
5 over 2
just like we did before we are going to get
the coefficient of x in this case the coefficient
of x is 2
Divide it by we divide it by 2
after dividing the coefficient of x by 2
We square the answer when you square the answer.
We are going to add it to that term
After adding it to that term then we
Get that term 5 over 2 n subtract the answer
work in this case. It's going to become 2
Into, it's going to become x squared minus
2 X so again negative 2 divided by 2 what
we get we get negative
1 negative 1 squared, what do you get?
Regarding 4 times negative 1 is positive 1
so we are positive 1 here
like that
Then we said last this 5 over 2 and
From that 5 over 2 that positive 1 we were
to be subtracting
That is this is how easy is giving us the
function of X which is equal to 0
So this is going to become 2 into now when
you simplify that quadratic expression it
is a perfect square
It's going to give us X minus
Minus 1 squared that's going to be plus. This
is over 1
You
After getting that we say that the function
X
X is minimum
When
X minus one is equal to zero therefore the
minimum value is
The minimum value of f of X is going to be
positive 3 and little piles
X is one because it because with X
minus one is equal to zero therefore u cos
when X is equal to positive one so
There's our that is how solve that question
and that is our answer
Let us do one more number by expressing 2
minus, 5x minus
3x squared in the form of a into that determine
the maximum value of f of X and
Stick the value of x at, which it occurs we
are going to use the same approach
But here this quadratic expression on got
a maximum value. Why because the coefficient
of x squared is a negative?
3 it's a negative so because the coefficient
of x squared is a negative
It means that this quadratic expression is
a maximum, so you are going to operate for
that maximum
But the worst within the same prison procedure
as before
So this is the quadratic equation it is 3?
So now we complete squares here still like
before we are going to get
The coefficient of x squared is 5 over 3 so
we shall get 5 over 3 and what shall we do
to 5 over 3
We shall divide it male 2, and then the answer
we get is squared so
It's going to become negative 3 into X square
plus 5 over 3x
so we get the coefficient of x pi over 3 divide
that by 2 is going to become 5 of
6 5 over 6 square it it's going to become
plus
25 over
36
So after adding this to that then we get this
and we say
2 over 3 minus that I'm sorry good. This is
going to become minus 2 over 3
minus 25 over 4 the 6 is giving us a function
of X like that
2x plus 5 over 6 squared
That will give us plus 49 over
Now of course the 36 and the kill this will
give us 12 when we divide them
It will give us the function of X so
We'll conclude Emma we say that the maximum
value
You
We want to
Investigate the relationship that exists between
roots and the coefficients of a quadratic
equation
When I talk about roots of the quadratic equation
in essence I'm talking about the solution
to a quadratic equation
If alpha and beta are roots of a quadratic
equation ax squared plus, BX plus C
It means that X minus alpha into X minus beta
is equal to zero
And also ax squared plus, BX plus C is equal
to zero have the same root X minus alpha
Into X minus beta
Is going to be equal to zero?
When you open brackets here, it's going to
become x squared
minus X beta
minus X alpha
plus alpha
Beta is giving us zero so we shall end up
with x squared minus X outside the brackets
into
theta plus alpha
Then plus alpha beta is equal to zero
Arlis
That's now erotic expression in X
good from expanding that
this
we relate that to this other equation since
they have in the symbols relating that to
Ax squared
Plus BX plus C is equal to zero when we
Divide through here by a
So that this goes with that so that the coefficient
of x squared is 1 we have that X square plus
B over X plus Sigma a is
equal to zero
So what do we see from there?
We see that the coefficient of x here and
the coefficient of x you find that this be
overhead
Is you cover into that and we also find that
this positive C over N
Is equal to that
from me I wishes you see that b + a disables
the sum of the roots is you managed to be
over a
and the product of the roots alpha times beta
is equivalent to
C over a so it is against that background
that we shall come up with this conclusion
For all quadratic equations, but generally
all quadratic equations of the form ax squared
Plus BX plus C is equal to 0 with roots alpha
and beta half the sum of dudes
alpha plus beta being equal to negative B
over a
alpha plus beta being negative B over a and
the products alpha beta being C over a a
Sequence that so now let's try some examples
230 thing write down the sum and product to
the following equations
Roman one Part A. We have
x squared minus 3x plus 2 now of course this
sum
Is going to be equal to negative B over a
In this case we have negative 3
The sum of a it's going to be negative in
actually positive 3 over L which is 1
So the sum is the sum the roots is 3
And then they product
Simply C over a which is going to be 2 over
1 which is 2 as simple as that let's look
at Part B
Part B is 4x squared plus 7x minus
3 is equal to 0
We want to find the sum and product of that
equation
this sum
Is going to be equal to negative B over a
which is going to be negative B B is 7 over?
A which is 4?
so this is the sum and
then the product is
Going to be C over a what is C C is
Negative 3 over a which is both
So the sum is negative 7 over 4 the product
is negative 3 over 4 as simple as that
So from this equation you will see that this
as
Long as the coefficient of x squared is 1
it is going to be x squared minus the sum
of the roots
Plus the product of the roots because the
sum is 3 the product is 2
It is the same here if we make X the coefficient
of x squared 1 in that phone
for example in that equation
It will be x squared is equal to 7x
squared plus if we divide through by 4 7 over
4 X minus 3 over 4
Is equal to 0 if the coefficient of x squared
is 1 you will see that the general form of
the equation?
X square
Minus the sum of the products plus
The protein I mean the sum of the roots plus
the product of zeros
so from that we come up with this general
equation of a quadratic equation that should
be at your fingertips the
general equation of a quadratic is x squared
minus
Sum of roots time of x X sum of roots left
right little head of X plus the product of
the roots
Now remember that you're going to use that
that x squared
Minus some that is this is for oh man 1
sum of roots
x plus product
Of roots
So in this case it's going to be x squared
Minus what's the sum of the roots here this
some history?
Plus what's the product the product is 4?
So that's the quadratic equation as simple
as that no man 2 is
Some is a third product is negative over 5
so it's going to be x squared
minus the sum of the roots these
apat X plus the product of the roots
Which is negative 2 over 5 so you end up with
x squared minus?
1/3 X
minus 2 over 5
as the quadratic equation let's do the last
one the last one is saying this sum is negative
two point five and
The product is is that if one point six
So it's still going to be x squared minus
the sum which is negative 225
X
Plus the product which is
Negative 126 so it's simply going to become
x squared plus 2.5 x
Minus 1.6. And what's our answer there?
Of course that's the quadratic equation, but
just remove though that this more point will
multiply through by T the equation becomes
x squared
10 x squared + 25 X
minus 16
that's the
quadratic equation
Express the following in terms of alpha plus
beta and alpha beta Roman 1 is alpha squared
plus beta squared
So we shall start straightaway with solving
Roman 1 forming alpha squared plus beta squared
in terms of this and that
We shall do we shall?
Get this from here
Alpha plus beta squared is going to be equal
to alpha squared plus 2 alpha beta
plus
Beta squared
If we make alpha and plus beta the subjects
that we come up with this expression which
is in our question
We shall end up with alpha squared plus beta
squared is going to be equal to
alpha plus beta
Squared minus 2 alpha beta and that's the
answer we have expressed this in
Terms of alpha plus beta which is required
in the question and alpha beta, which is that
so that's our answer
Let's move on to Rome and to
Rome and to is
alpha minus beta
square
so alpha minus beta squared
That is going to be
alpha squared
Minus 2 alpha beta plus beta
Square this is going to become alpha squared
plus
beta squared minus 2 alpha beta
But we already found what alpha and beta is
We already know that alpha squared plus
beta squared
Is this and it's equal to that so we shall
substitute for this and just bring that expression
there
so in essence here, that's going to be equal
to
in brackets alpha plus beta
squared minus 2 alpha beta then minus 2 alpha
beta
So our answer will be alpha plus beta in brackets
squared minus 4 Alpha we
That's the answer let's do Roman 3
roman 3 is
alpha tubed +
vttitude
Now for alpha tube plus beta tube. We are
going to use the same technique. We used before
we shall use
We shall get this expression from alpha plus
beta all in brackets to the power 3 that is
the same as saying
alpha plus beta
Squared times alpha plus beta like that plus
2 alpha beta o
Plus beta squared all this is multiplied by
alpha plus beta
Like that so we open these brackets these
times that is going to be alpha cubed
Veta let's rewrite this down
It's going to be equal to alpha tubed plus
beta alpha squared plus
2 alpha squared beta plus 2 alpha beta squared
plus
alpha beta squared plus beta
cubed
So from that expression we make alpha chipped
and beta tube the subject of the formula
So there is topologies which I'll shift it
to the other side, but before we do that
I think we can simplify this a bit further
to make it a bit shorter
so alpha plus beta
cubed is going to give us
Let's call this alpha cubed
So that we can who mean it's alpha cubed
Plus let's pull out the common terms is going
to be three alpha beta into that's going to
be alpha plus beta
plus beta
Tube is going to give us alpha plus beta
cubed
so from there
We make alpha tube and beta to be subject
to the formula and the rest goes this side
so it will
remain with alpha plus beta
Cubed this one this expression crosses the
equal signs it comes this way it becomes minus
3 alpha
beta
into alpha plus beta
Is giving us alpha cubed plus b cubed so?
Just rearranging that with me that alpha cubed
Plus beta cubed is giving us
alpha plus beta oh
This is the chip this outside here
alpha plus beta in brackets cubed minus 3
alpha beta
into alpha plus beta
And that is our answer
In essence we have expressed alpha cubed plus
beta tube in terms of alpha plus beta and
alpha beta
Then they works of the equation 2x squared
Minus 4x plus 5 is equal to 0 our alpha and
beta find the values of 1 over alpha plus
1 over beta
Prodigy's, but we and this part c and so on
and so on so
How do we go about this kind of question they
are telling that they've given us an equation
the equation is?
2x squared minus 4x
plus 5 is equal to 0 and they're telling us
the roots of this equation are
alpha and
Beta
So it means in these roots?
the some of these roots
Is supposed to give us a certain number the
product of this root is supposed to give us
a certain number
but from our earlier videos we
Explored that the general equation of all
quadratic equations is going to always be
x squared
So you see that this quadratic equation is
in this expression so it means that the sum
of roots should be alpha plus beta
Here whatever figure is there and the product
of the thing there? They don't is alpha beta
from our previous videos we came up with a
conclusion that
Asserted that generally all quadratic equations
in that form ax squared plus, BX plus C is
equal to 0 with roots alpha plus beta
How there are some of the roots being it with
negative B over a and?
The product of the roots, which is alpha beta
being equal to C, or a so in this case
with this equation here, 2x squared minus
4x plus
5 being equal to 0 we know here that the sum
If we made X the coefficient of x squared
will be 1 we end up with x squared
minus
4 over 2x
Plus 5 over 2 is given give us
From the general expression we know that the
coefficient of area is 1
Our value of B. Is that how Heidi of C is
5 over 2 positive 5 over 2 and
When we compare that with the general equation
of the quadratic equation the general expression
which is ax squared plus?
BX plus C. You will find that the sum of the
roots
Going to be equal to negative V over a that's
the sum of roots
negative B over a so
if I sum the sum of the roots is negative
B over a
it means that it's going to be what is earning
that will be our negative B is a
negative value B is negative 4 over 2 which
Is going to give us positive 4 over 2 which
is 2?
And what about the products of the root?
The virus is going to be
C over a
And in this case is going to be five over
two
That's the product of the rules, but then
we have been told that verbs are alpha and
beta
So it means that the sum of the roots?
Which is alpha plus beta is going to be equal
to 2
And the product of the roots, which is alpha
times beta is going to be equal to 5 over
3
This is the sum of their roots
This is the product of the errors so that
being extrapolated from the question you can
go ahead and start answering the questions
They're telling us to find the values or values
of Part A. They are telling us 1 over alpha
plus 1 over beta
Now for us to find the value 1 over alpha
plus 1 over beta. We are simply supposed to
calculate this expression
arithmetic in such a way
that we
Make it convert it into this form into forms
of alpha plus beta and then alpha beta
at the end of the day we shall simply just
Substitute in the values of alpha plus beta
and alpha beta, and then we shall get our
expression so here
We shall go ahead and find the LCM which is
alpha beta alpha beta divided by that that
is beta
up
plus
alpha
So our answer is beta plus alpha is to
Divide that by alpha beta which is 5 over
2 so it's going to be 4 over 5 but B is
They're asking us to find alpha minus beta
squared
Now the expression for alpha minus beta squared
we already produce. They are explained in
in our earlier calculations
the expression for alpha alpha minus beta
squared we ended up with
alpha plus VJ
Squared minus 4 alpha beta we got this expression
in our first example. We shall not repopulate
it we shall not catch it again
How do you rate it again? So this is going
to be 2?
squared minus 4
times
5 over
It will be four - by two ions by two twice
two times five is ten
That is 4 - 10, and we shall end up with negative
6, and that's our answer
That is our Part B. So going to Part C
Our Part C is asking us to find the value
of 1 over
alpha plus
1
plus 1 over
beta plus 1
Now 1 over alpha plus 1 plus 1 over beta plus
1 is going to be equal to we find the LCM
of these
It's going to be alpha plus 1
into beta plus 1
here we are trying to make sure that we convert
this expression in terms of alpha plus beta
and alpha beta so that we
End up substituting so this divided by that
Is going to give us beta plus 1 so we have
beta plus 1 out there plus?
This divided by that will give us alpha plus
1 alpha plus 1 times 1 is going to give us
alpha plus 1
So we shall end up instead with
This Plus that we end up with beta plus alpha
plus 1 plus 1 is 2
Divide all that by these things that is alpha
beta
Plus these things that is alpha plus 1 times
beta is
beta then 1 times 1 is 1 plus
1
So the answer we shall get here will be
beta plus alpha
is to
Remember alpha plus beta is to the sum of
the roots is 2 so it's going to be 2 plus
2
Divide that by alpha beta
That's the product of the roots and according
the other we find out the product of the roots
is 5 over 2
plus
alpha plus beta alpha plus beta in this case
is 2
in plus 1 we find that we are able to manipulate
the
expressions for
alpha and beta
We make sure that we convert them to make
sure that
Alpha the expressions we see here or any other
expression that are given
You first convert them in the form of alpha
plus beta
Because you know that the value of alpha plus
beta is 2 and
You also express them in form of alpha beta
because you know that they of the value of
alpha times beta
Is the product of the roots which you have
been given and that's how we find our?
Expressions our answers as 8 of 11 the values
of the expressions
The roots of the equation x squared minus
2x plus 3 is equal to 0 our alpha and beta
Find the equations whose roots are alpha alpha
beta
comma beta over alpha so now from the equation
x squared minus 2x plus 3 is equal to 0
We shall find that we will be told that the
roots are alpha and beta
So the sum of the root is alpha plus beta
the product is alpha beta so the sum of the
roots alpha plus beta
are still be going to be equal to negative
B over a
These are terms that are within this quadratic
equation that has been given to us within
the question
so in this case our value of B
Is negative 2, which is that our value of
a is positive?
1 the coefficient of x squared so of the sum
of the roots is going to be equal to 2
Then the product of the roots is alpha beta
Which is still going to be able to see over
our your C for my equation is 3?
And our value of a is still 1 so the product
of the roots alpha beta is going to be
3 over
1 and
our 3 over 1 is still giving us 3
So in conclusion from our equation that has
been given to us in the question
We find that the sum of the roots, which is
alpha plus beta is going to be equal to 2
and
Alpha beta, which is the product is going
to be equal to 3 so from that this knowledge?
We are going to now start answering our question
Our part is telling us to find in the equation
the quadratic equation whose roots are alpha
over beta comma beta over alpha
Now we are going to find the sum of these
roots first
But we are supposed to manipulate these two
roots in such a way that
The expressions we get are in the form of
alpha plus beta and then alpha beta so that
in the end we are able to
substitute for alpha plus beta as 2 we substitute
for alpha beta as 3 so that we are able to
get
the
Sum and product of them respect roots like
we shall sing about conditions so our part
is going to be alpha
over beta comma beta over alpha
So the sum of the roots is going to be
That plus that
LCM of beta and alpha is alpha beta alpha
beta divide that by beta. You're going to
get alpha alpha times
Alpha is going to give you alpha squared
Then again alpha beta divided by alpha is
going to give you beta beta times beta is
going to give us beta squared
So we come here from our previous videos.
We were already derived
We already know how this comes about
Maybe just to repeat it a friend who digs
the sidewalk here is alpha plus beta
We get it from alpha plus beta in brackets
squared is going to be equal to alpha squared
plus
2 alpha beta plus beta squared when we make
alpha and beta squared alpha Square
And we describe the subject of the formula
so that we obtain this expression
We end up with alpha plus beta in brackets
squared
Minus 2 alpha beta so this expression we substituted
for this here
So end up with alpha plus beta is 2 so it
is 2 squared minus
2 times alpha beta
We have beta there
So it's 2 my 2 minus 2 alpha beta
So it's going to be 2 times alpha beta
Which is 3?
So 2 squared is going to be 4 and 2 times
3 is 6 so 4 minus 6 is negative
2 divided by 2 you end up with negative 1
so the sum of the roots here is negative 1
after finding the sum of the roots as negative
1 you
Go ahead and find the product of the product
of the roots. It means you're going to multiply
the roots together
So here the product of the roots give us these
times that we have alpha beta
divided by alpha beta which is going to be
3 over 3
3 divided by 3 will be first one
So after finding the sum of the roots as negative
1 the product of the roots as 1
We come ahead and record and substitute in
that expression
X squared minus sum of roots plus per our
roots in this case is going to become x squared
minus
The sum of the roots is negative
1x plus the product of the roots which is
1
so our
equation alpha reading Persian is going to
become x squared these things that is plus
X plus 1
That is the quadratic equation we get in that
case
so we shall go ahead and do our part b
but b of this question is
find the equation I mean find the
quadratic equation when you've been given
these two roots the roots here the first one
is alpha minus beta
Comma and the second is beta minus alpha
We are going to do this number like we did
in the previous one. So we shall go straight
ahead and find
the sum of the
The rule of these words will be alpha minus
beta
plus beta
minus alpha
that's going to be equal to now we find that
alpha minus alpha will be 0
Negative beta plus beta will be 0 so it means
here the sum is 0
So let's find now the product
Productive the roots, we are going to multiply
those two terms together
So it's going to become alpha minus beta
Times beta minus alpha that's going to be
these times that that is alpha beta
these times that is
minus alpha squared these times that is
Minus beta squared then these times that is
positive alpha beta so here
What do we have we have this and that that's
going to be 2 alpha beta
Then this is always the minus in brackets
of a squared plus
beta squared
Wow so 2 alpha beta minus
What is the expression for alpha squared plus
beta squared
we already found that in our previous calculations
alpha squared plus beta squared is equivalent
to
Alpha plus beta
Squared minus 2 alpha
Beta 2 alpha beta open bracket. It's great
with minus
alpha plus beta
squared these times that is plus 2 alpha beta
So we shall end up with
This plus that is going to give us full alpha
beta minus this
alpha plus beta
Squared so our answer here is going to be
4 times
What is in alpha times beta is 3 because?
according to our
General equation given to us in the question
the product of the roots is 3 so it's going
to be 4 times 3 minus
According to the equation. We've been given
in the question the sum of the roots was 2
so it's going to be 2 squared so
We shall our answer here. I shall end up with
4 times 3 4 times 3 is simply 12
Minus 4 and our answer is 8 so here the product
of the roots is going to be 8
So he start formulating our equation by simply
saying that
Simply equal to x squared minus some of the
roots is going to be
required the sum of the roots are
0x
Plus the product of the roots which is 8?
so of course our quadratic equation will end
up being x squared
This term will die so it's going to be x squared
Let's call it minus 0 plus 8 so we shall end
up. It's going to become x squared plus 8
we are supposed to find the sum and
The product of these two roots, so let's first
find the sum of the roots
Is going to be
1 over
alpha plus 2
squared plus 1 over
beta plus 2
squared that's going to be equal to
alpha plus 2
squared
That's more like now the LCM beta plus 2 squared
this divided by that
We are going to get this screen with beta
plus 2 squared plus this and that we are going
to
This will go you remain with alpha plus 2
squared that's
Going to be 2 squared
Plus 4 beta
Plus 4
Then we expand this as well, it's going to
be plus
Alpha squared plus 4 alpha
plus 4
Divide all that by now. This is the same as
saying
Alpha plus 2
Into beta plus 2 all this is squared
so continuing our calculation
This is going to give us
You
Opening those brackets and ending up with
that kind of arrangement
We shall end up with
we know the expression for beta squared plus
alpha squared from our idea calculations it
gave us
alpha plus beta
Squared minus 2 alpha beta this is equivalent
to that
We derived it before we shall not to do that
again plus
So we shall continue plus 4 into beta plus
alpha plus 8 plus
4 into beta plus alpha plus 8
divide all that by
alpha beta plus 2 into alpha plus beta plus
4
All that is squared so I think from here.
We can see that everything has been converted
It is in the form of alpha plus beta and alpha
beta alpha plus beta alpha beta so shall congest
substitute we know that the sum of Arts according
to the question is 2
So this is going to be 2 squared minus 2 times
3
plus 4 times 2
Plus 8 divide all that by alpha beta, but
the Alpha Beta Alpha Beta is going to be
3 plus 2 into alpha plus beta is 2 plus 4
all that is
squared so
our answer there is going to become 14 over
1
To 1 after finding the sum of the roots. We'll
go to the product of the roots so product
So that is going to give us one
Over these times that lets me down here. That's
the product
So it's going to give us these times that
is going to give us 1 over
1 over alpha plus 2 squared times beta which
is
plus 2 squared 1 over
Put all that in brackets alpha plus 2
0:59:43.279,0:59:46.209
That is one bracket the other is beta plus
2 all this is squared so it's going to become
1 over
Put that in brackets the expression is alpha
beta
Just like we did before plus 2 into alpha
plus beta plus
4 all that is squared as in alpha bit already
from alpha times beta
Then plus alpha times 2 which is 2 alpha plus?
2 beta which is to beta
Plus 4 2 times 2 which is 4?
So it shall end up here with 1 over
What is alpha beta of a beta is 3?
plus 2 into alpha plus beta is 2
Plus 4 all this is squared. We end up with
1 over 1 - 1
Make that
1 over 1 - 1 is the answer we end up with
So meaning here these are the product of the
roots
So the product of violets is going to be one
over one to one so in essence the final equation
is
Going to simply be x squared
Minus the sum of the roots the sum of the
roots we got was 14 over 1 to 1
So it's going to be 14 over 1
to 1 X
Plus the the product of the roots which is
going to be 1 over 1 to 1, so
That's the quadratic equation we get
Equations with common roots are simply equations
that shear a root
Let's look at this example given that the
equations
x squared minus 2x plus a is equal to 0 and
2x squared minus 5x plus
B is equal to 0 have a common root by igniting
X show that the root is B minus 2a
So to show that this root is equal to B minus
2a. We simply
first write those two equations and
the two equations in this case are x squared
minus 2x plus a is equal to 0 and
then 2x squared minus 5x plus 3 is equal to
0
so for us to eliminate x squared it means
that you're supposed to make sure that this
value all the
Coefficient of x squared is the same so that
we subtract the 2 it is 0?
so
We multiply
This coefficient of x here into the whole
of the first equation
So that in so doing we make to 8x squared
also have a coefficient of 2
So here what we shall do
We shall get these two and multiply through
two times x squared is 2x squared 2 times
negative
2x is negative 4x then 2 times a is 2 n is
equal to 0 then the second equation
Simply remains the way it is
so we subtract this first equation - the second
one 2x squared minus 2x gives us 0
negative 4x minus
negative 5x of course this becomes a plus
negative 4 plus 5 gives us positive x
then 2 a minus B. Very simple remain as 2
K minus B is equal to 0
So we find our root which is X is?
Going to be B over a and. This is the common
room so here
We finished
but which telling us to show that this root
is big - too late and
Indeed we have shown that the common root
between these two is b minus 2a
so going to our next
Part of the question that is Part B, but B
is telling us to find
Find the expression for the other root of
the equation so in this question we have two
equations here, we have the first equation
giving us
The first equation according to our question
as you can see is this
Our second equation is not and we have known
that
one both these equations
share a truth B minus a
so
We are supposed to find the expression of
the other root
It means that this equation has this root
plus another word this equation here has got
this root
plus another root
So we are supposed to find the expression
of that other word so we shall begin with
our Part B of the question
We shall begin with all of the equations x
squared minus 2x plus a
One of the roots of this equation is B minus
2 a so now what is the other roots?
Let the other root B lambda
So if we said let the other loop root B lambda.
We look at some of roots according from half
past
videos we say that some of which is going
to be equal to
Negative B over a and we know that negative
B
Over a is the same as negative into what is
B according to the equation the value of B
is negative 2, so?
It's going to be negative 2 over a our value
is the coefficient of x squared which is 1
so negative times negative is positive
and so you end up with positive 2 so the sum
of the roots is 2
So we shall we simply add the sum of the roots
we know that
Lambda which is the root? We are looking for
plus the other common root, which we have
already
proven which is in essence be minus 2 a
not b minus a by B minus 2 a
Is going to be equal to 2 since lambda is
going to be 2 minus in brackets B minus
2 a
B minus 2 a
So we shall get our answer as to - be
Negative times negative is positive. So it's
going to be plus K
So this is the expression for?
The adult for that equation then looking at
the second equation in our question
The second dreadful equation as 2x squared
minus 5x plus B is equal to zero
So let the other root be lamp alpha
we know that the sum of roots is equal to
negative B over a the some roots of the quadratic
equation out of order is
Going to be both negative B over a so in this
case. What is negative B
What is B B is negative 5 so we substitute
a is going to be negative?
B which is negative 5 over a what is the value
of a the coefficient of x squared it is 2?
So it's negative 5 over 2 negative times negative
will give us positive
So it's going to become 5 over 2 so this is
the sum of those two roots for this equation
So we simply go and say since this up with
good the sum of roots as 5 over 2. We'll go
ahead and say that
that
That should be minus. 2a, which is the component
that is shared by these two equations?
Plus alpha the root. We are looking for is
going to be equal to 5 over 3 and
Alpha is going to be called 5 over 2 minus
Vista
I mean this term will be minus a currency
sign comes there
When it crosses the cosine becomes negative
this way
So it's 5 over 2 minus 2 B minus 2 it. Pop
in brackets here. It is negative 1 times B
is negative a negative
times negative 2 a is going to be plus 2,
so it's alpha is going to be 5 over 2
minus V plus 2 a and that's going to be the
expression for the other are we
So let's let's look at the another question
here
Given that
2x squared plus 3x plus M is equal to 0 and
3x squared plus X minus 2 M is equal to 0
have the common root find the possible values
of M so these
two equations
This equation and that equation have a common
root
And they're asking us to find the pursuit
with the possible values of M. If these two
equations are having the common root
We shall go ahead and try to find that root
We shall approach this like though. We are
doing in our previous
calculation
so we shall write down this equation is that
they're
2x squared plus 3x plus
M is equal to 0 they may have that equation
which is 3x squared plus X minus 2 M is equal
to 0
Then we eliminate X square
eliminating X squared means
This the coefficient of x square must be the
same, but since in this case they are not
the same
We shall get three multiplies in the first
equation
Get these two
multiplied through in the second equation
So that at the end of the day the coefficient
subjects are the same and when we subtract
them
We are able to get here so here 3 times 2
is 6 3 times 3 is 9 3
times M is 3 M is equal to 3 times 0 which
is 0
Then in the second equation it is 3 times
2 which is 6 X?
2 times X is 2x
Then we have 2 times negative 2 is going to
be much negative 4 in
Giving us 2 times 0 giving us 0 so we have
this equation and that equation
So we subtract this equation this - that equation
so 6x minus 6x squared will give us
nine X minus
Two X will give us seven x
3m
minus negative, 4m these two
Minuses become a positive so it becomes three
M last for him is giving us seven M. Is equal
to 0-0 in velocity
So this 7m crosses the equal signs to come
this woman in this process the equal signs
We end up with 7x on this time divided by
seven is equal to negative seven M divide
that by 7
we
Cancel out the sevens you mainly deliver value
of X as negative M. That's our value of x
after finding our values X as negative M
We are now going to have to substitute this
negative M into any of those equations that
we've been given in the question
So that we are able to find the range of values
for him the possible values of M so in this
case
It's going to become three we get just the
second equation 3x squared plus X minus two
M is equal to zero
So where there is X we substitute for?
Negative M and in this case it is going to
be 3 times negative M. Squared plus X
M minus 2 M is equal to zero
So we'll continue without alkylation and say
that this is going to become three negative
M times negative M is
Going to be positive M squared plus
This is negative M
- - M is equal to zero
Of course this negative and positive will
give you the negative value there
So it becomes three M squared minus M minus
two M
0 to 0 and
From there. We shall end up with
3 M squared negative x minus 2 M is going
to be negative 3 M
Is going to be equal to 0?
So when we pull out the common factor here
it is 3 M into M minus 1 is equal to 0?
So here Ava
3 M is equal to 0 for
M minus 1 is equal to 0 therefore the value
of M here will either be 0 for
The value of M. There is going to be equal
to 1
So we answered the question the question required
us to find the possible values of M
And now we have put in them M is a 0 for M
is
negative 1
