Hello everyone, welcome back to yet another video on
quadratic equations.
In the last video, the first video of quadratic equations,
we had seen the basics of equations.
What is an equation? What is a quadratic equation?
Then, we had seen the solution, general form, how to write
the solution, how to find the roots, we had seen the
formula.
And we had seen the nature of the roots.
What type of roots will we get?
They depend on the value of discriminant.
We had seen all this till the last video.
And we had also asked this question.
So, the slide has stopped there.
Then, I had told you that we will see more questions in
this.
So, let's see the remaining questions.
So, see this.
See, nature of roots.
We had studied the nature of roots.
This is the question about that.
So, the roots of the equation is 3x square minus 12x plus
10 is equal to 0.
And, what type of roots are there in this equation?
Are they real, rational, and unequal, complex, irrational,
real, equal, and so on.
So, to know this, first of all, what we have to find out?
We have to find out discriminant D.
What was the formula of discriminant?
Whatever was inside the root, that is b square minus 4ac.
So, first of all, write a, b, c.
What did we get from this? a coefficient of x square, that
is 3.
b coefficient of x, that is minus 12.
c, constant term with sign, that is plus 10, so 10.
So, we got the value of a, b, c.
Then, put b square would be 12 square, that is 144 minus 4.
a, 3 and c, 10.
What is this? 144 minus 12, 120.
So, what is this value? 24, that is positive number.
Greater than 0, which is positive number.
So, when our discriminant was positive, then what type
would be this?
If we see the solution, then minus b plus minus under root
24 divided by 2a.
Now, what is the value of b?
So, minus 12, that would be plus 12.
Plus minus root 24 divided by 2 into 3.
Now, tell me, this root 24, is it a perfect square of any
number?
Obviously, it is not a perfect square of any number.
So, whatever we will write, what will be the final answer?
Our answer will be in the root.
Now, if it is in the root, then let us factorize it.
So, how can we write 24? 2 into 12, then 2, 6, 6.
Then 2, 3 and 3.
We can write this as 8 into 3.
We can write 8 as 2 into 2, 2 cube into 3.
So, we can write root 24 as 2 square into 2 into 2, that is
6.
We can write it like this.
Then 2 square, this will come out.
We can take 2 as common.
So, what can we write by solving this?
We can write this as 12 plus minus 2 root 6 divided by 6.
Now, if we take 2 as common in the whole equation.
If we take it as 2, then if we divide 2 by 6, what will be
left? 6, if we take it as 12, then 6.
If we take it as 12, then 2 will get cancelled out.
Plus minus root 6.
And if we divide it by 6, what will be left? 3.
We have cancelled out 2 as common.
In the denominator and numerator also, we have the final
answer.
So, what are the two solutions? 6 plus root 6 divided by 3.
And 6 minus root 6 divided by 3.
Now, this root 6, what number will come?
Would it be rational or irrational?
We have non-square term for root 6.
So, obviously, what will come?
Real will come.
But, see this C part with real.
It should have come equal with real in the part.
But, here it will come unequal.
In one, 6 plus root 6.
In another, 6 minus.
So, this will come unequal.
Real will come, but unequal will come.
So, what is this option?
We have discard.
Similarly, this equal option also got discarded.
Because it is not equal.
Now, what is complex?
B part is complex.
Is this complex? Imaginary part is coming in this.
So, negative part is not coming in the root.
So, it is not imaginary part.
So, we do not have this complex also.
They are real.
Now, we have two parts left.
D and A.
Now, this root 6 will give you irrational term.
Because it will not come.
So, it will give irrational term.
That is why this answer will come.
D part.
That is irrational and unequal.
So, it is there.
What happens in rational?
When it comes out of the root.
If this whole root would have come out.
Then we would have got a number in P by Q form.
That would have become our rational number.
But, now it is not happening.
So, what kind of number is this? Irrational and unequal.
So, irrational and unequal.
So, we have discarded this nature of roots.
We are taking so much time to do this and explain this.
But, when you solve this, you will get the answer in a
jiffy.
And, this becomes span of time.
Next question.
If the root of the quadratic equation are 20 and minus 7.
So, value of X is 20 and minus 7.
I told you that you can get the solution from equation.
And, similarly you can do back way.
You can get back from root to equation.
How? For that, you have taken first root of X1 and second
root of X2. What you have to do?
You have to find the sum of these two.
And, you have to find the product of these two.
So, what will be the sum?
Sum that is X1 plus X2.
That is 20 plus minus 7.
So, plus minus minus.
20 minus 7 would be 13.
So, you have the sum of 13.
The sum should be equal to minus B by A.
And, minus B by A should be equal to 30.
So, it should be equal to minus B by A.
Now, we are going to find the product of these two.
Product will be 20 into minus 7.
That is minus 140.
And, it should be equal to C by A.
It should be equal to C by A.
So, you have minus B by A and C by A.
If you have an equation of this type.
In which, the value of X is 1.
So, in all, X has coefficient 1.
So, what type of equation is this? This is the type of
equation.
Here, what was there?
Minus sum X plus product is equal to 0.
Do you remember, I had written this? When I had divided the
whole equation by A.
So, you had this.
So, I put the value of sum.
The value of sum is 13.
So, what will happen?
Minus 13.
So, X square minus 13 X.
Plus, what is the value of product? Minus 140.
So, minus 140 is equal to 0.
Solve and see which one is coming.
The last one should be minus 140.
So, it can be either this or this.
How should 13 be?
It will go in negative.
So, this will be our C option.
Because, this should be minus B by A.
So, minus sign should be there.
Understood, how to go backwards?
Check this.
So, you don't have to do this.
Just take sum and product.
Tell and compare which one is coming.
With the sign.
Sign is very important.
Because, coefficient is same for all.
In all, coefficient of X is 1.
X square is 1.
X of 13 is 13.
And, constant term is 140.
But, what is the sign in front of them?
With that, our equation is changing completely.
Understood.
So, this question is also done.
Now, next question.
What is given in this?
In this, instead of giving value of roots, relation is
given in roots.
And, that relation is given in ratio.
So, what is this?
Find the value of B. If you don't know the ratio, then
please go to the video on the topic of ratio and know the
meaning of ratio.
Then you can solve the equation.
I guess everyone knows the ratio if you are studying the
quants and aptitude.
So you should know that if you have two equations, write
the equation first.
2x2-5x plus B. This is your equation.
This is equal to 0.
What is the constant term given to you this time? It is B.
And the coefficient of x should have been, if I put the
value of A, B, C, then the value of A becomes 2.
The value of B becomes minus 5.
And the place of C is given to you as B.
Don't get confused. This is the equation B and this is our
general form B.
So the roots are given to you in the ratio 2 is to 3.
So let common ratio be x.
So if you have x1, x1 is 2x.
So what will x2 become? Simply multiply 2 by x, so x2
becomes 3x.
The second one is 3, which is after the ratio.
So your second root is 3x.
So you have two roots x1 and x2.
Now you have to find the value of B.
What is the value of B?
Constant term, constant and constant B.
In this, the coefficient, what should you do first?
No need to make coefficient 1.
No, what is the last one?
Product of roots.
What happened? 2x into 3x.
What is this equal to?
C by A.
What is the value of C? B.
What is the value of A? 2.
So what is this equal to?
B by 2.
Solve this.
What do you have?
2 into 2 is 4, 4 into 3 is 12.
x2 is equal to B.
The value of B is 12x2.
Don't take common ratio as x.
Because this is an equation in x.
You take z.
So 3z and z will come.
You will get z2.
Why am I taking z?
Just because this is x and z.
This is x variable and this is unknown variable.
Don't get confused with z.
That's why I took z.
To find out the value of B, what should you have?
The value of z should be there.
Only then you will know the value of B.
For the value of B, use the formula of sum.
What will be the sum? 2z plus 3z.
What should it be equal to?
Minus B by A.
Put the value.
What will be the like terms? 5z is equal to minus B.
That is minus of B.
The value of B is minus 5.
Divide it by 2.
This is plus.
5 cancels with 5.
What is the value of z? 2.
Put the value of z here.
The value of B will come.
That is B is equal to 12 into 1 by 2 whole square.
That is 1 by 4.
1 by 3.
What is the value of B? 3.
You have seen how to find the value of B using the formula
of sum.
Next.
The sum of the squares of 2 consecutive positive integers
exceeds their product by 91.
Find the integers.
We have given the integers.
Now we can write 2 consecutive integers.
We can suppose let.
If we say integers, natural numbers, whole numbers, then we
can say let.
Because we know that if 1 integer is 2, then 3 will come.
In the gap of 1 and 1, all the integers remain line wise.
We know this.
But if someone else says rational number, real number, then
we cannot say consecutive.
We cannot say which one will come after 1.
That is why positive integers are mentioned in this.
2 consecutive positive integers.
What is there in positive integers?
Whole numbers.
Because there is no zero.
So positive whole numbers.
Positive natural numbers.
That is whole numbers.
Let us suppose 1 is Z.
What will happen to the other integers? If 1 is Z, then its
consecutive.
Its front or back can be any.
What will be the gap in that? 1.
So next automatically we will have Z plus 1.
Or you can take Z minus 1.
Or Z plus 1.
Because we can have its front or back.
What will be the answer?
Both numbers will be same.
So you took 1 as Z.
And you took 2 as Z minus 1.
So the sum of the squares of 2 consecutive positive numbers.
The sum.
Whose sum?
Square.
So Z square plus Z plus 1 whole square.
This should be equal to what?
Exceeds their product.
How big is this bigger than its product?
This is bigger than its product.
So Z becomes their product.
This is bigger than their product.
This is 91.
Means its value.
Is bigger than this.
This is bigger.
How big is this? 91.
If we add 91 more in this.
Then we will get this.
So how will I write this in equation?
This is that.
Means inequality has come.
Means greater than sign has come.
So we need that equation.
So what will we do?
I said exceed by 91.
That is this is bigger than this.
91 steps more.
So if we add 91 more in this.
Then we will get this.
Or subtract 91 from this.
Then we will get this.
So what to do?
Simply add here.
Plus 91.
And what will happen then?
Then these two things will be equal.
Then find the integers.
Now in this what you find out? Integers.
So in this, Roots did not work.
Means equation did not work.
But what can we do?
We will see how it will work.
Now it is in power 2.
Let us solve this first.
Look at this.
This will remain as it is.
Open this.
What will be the identity?
A plus B whole square identity.
That is A square.
This is A and this is B.
So A square that is Z square.
Plus B square that is 1 square.
Plus.
Because plus is 2AB.
That is 2Z into 1.
That is 2Z.
This is your square.
Multiply Z individually.
Distribute it.
Z into Z is Z square.
Z into 1 is Z.
Plus 91.
Now you have this side and this side.
So this Z and this Z will cancel out. Because this will
become minus Z.
So what is left?
Solve like terms together.
One term of square is left.
Z square.
One term of Z is 2Z.
Second term is Z.
So this Z will be taken here.
2Z will become minus Z.
So what will be left?
Plus Z.
Now this 1 and this 91.
Solve them together.
Minus 91 will be left.
Minus 90 is equal to 0.
So your quadratic equation is like this.
Z square.
Z square plus Z.
Minus 90 is equal to 0.
In this unknown variable, you took X and I took Z.
In Z equation is like this.
So highest degree is 2.
So quadratic equation is like this.
Now we have to find the integers.
Which are those integers?
Obviously Z and Z plus 1.
So we have to find Z value.
What is Z?
Its solution.
So how will be the value of solution?
Z is equal to minus B plus minus under root B square minus
4AC divided by 2A.
So what is the value of Z?
Z's coefficient is 1.
That is minus 1 plus minus under root B square 1 square
minus 4 into
A. A is 1.
What is C?
Minus 90.
Minus 90 divided by 2 into one that is 2.
So, this comes after solving.
Minus 1 plus minus under root minus minus plus.
So, 1 plus 4 into 90 that would be 360 divided by 360.
Solve this as 361.
361. Whose number lies between 4 and 100? It is smaller
than 4 and bigger than 100.
So, it is bigger than 10.
So, it lies between 10 and 20.
Now, see what is the square of 15? 225.
It is smaller
than 20.
So, it lies between 15 and 20.
If it is a perfect square.
So, what should come between 15 and 20? 1.
Whose square will 1 come in the end? 19.
9 times 9 is 81.
So, once you check 19.
19 into 19.
That is 9 into 9 is 81.
9 times 9 is 17.
It will cross.
This is 19.
1, 6, 1.
361. Whose square is this? This is the square of 19.
So, if we write this as square, then it will cut the root
from square.
So, this comes as minus 1 plus minus 19 divided by 2.
So, first minus 1 plus 19.
Divide by 2.
This would be your 18 by 2.
That is 9.
And what is the second value you have? Minus 1 minus 19 by
2.
That is minus 20 by 2.
Minus 10.
So, two values have come.
Your 9 and minus 10.
So, which one will you put? Two values of Z have come.
9 and minus 10.
So, can we write Z value as minus 10? Can we take it? See,
here in the question you were already mentioned two
consecutive
positive integers.
So, this is not positive.
Minus 10.
So, what will we do with this value?
We will discard this value.
So, what is the value left? Only 9.
So, value of Z has come to us.
9. So, what is Z plus 1? 10.
So, 9 and 10 are two consecutive positive integers
which satisfy the property mentioned in the question.
It was a simple question.
There was no hard and fast rule in this.
You just solved it.
You should know how to make this equation in the word
problem.
You should know how to make the equation and then how to
solve it.
You just have to apply the formula.
As such.
Now, next question.
Read this question.
If a and b are the roots of the equation, x square minus 9x
plus 20 is equal to 0.
Then what we have to find out? a square plus b square plus
2ab.
It is simple.
It is the simplest.
What you have to do? This is just to scare you.
a square plus b square plus ab.
So that
you don't remember any identity. You just have to find out
the two roots of the equation.
The name of the roots was x1 and x2. In this equation, the
name of the roots was a and b.
Then the values of a and b will come. Put them in this
equation and find out the answer.
Solve it quickly. Till then I will solve it here. So, a and
b. What will come? a and b.
What is the value of x? Minus b that is plus 9 plus minus
under root.
b square that is 81 minus 4ac.
4ac that is 80.
4 into 20.
2a that is 2.
So, 9 plus minus 1 divided by 2.
10 divided by 2 is 5. 5 and 4. Put the values. Take any a
or b because it is symmetric.
It is the same value. So, we don't have to give any
relation as such that a should be small and b should be big.
Put any value because it doesn't make any difference. So,
what is this? We take a as 5 and b as 4.
So, 25a square plus 16 plus 5 into that is 20.
So, what will come after solving this? 11, 1, 5, 6, 5, 6,
61.
What will be your answer? 61 that is option number C.
It was easy.
You do it yourself.
Last question of today's video is
Find the quadratic equation whose roots are the reciprocal
of the roots of this equation.
2x square plus 5x plus 3 is equal to 0. This is a new type
of question.
You have been asked to find the quadratic equation whose
roots are the reciprocal of the roots of this equation.
Suppose the roots of this equation are x1 and x2. So, the
roots of this equation should be 1 by x1 and 1 by x2.
Now, the values of x1 and x2 which we used to get, whose
values did they belong to?
They belonged to the formula of x is equal to, right?
They belonged to the value of x. We used to name them as x1
and x2.
But the values belonged to x.
So, if the values of x belong to this formula, then what
will happen to the reciprocal?
The solution of that should be 1 by x.
So, what you have to do is, in this equation, which is your
main equation, 2x square plus 5x plus 3 is equal to 0.
In this equation, you have to replace x with x1 by x.
So, you will replace x with 1 by x.
So, put it. This will become 2 1 by x square plus 5 1 by x.
3 does not have x, so 3 is equal to 0. 0 is also as it is.
Now, what you have to do is, 2 by x square plus 5 by x plus
3 is equal to 0.
How do we need the equation? We need it above.
We need the general form of the quadratic.
x square plus ax square plus bx plus c type.
So, you complete the whole equation with what?
Take this LCM, x square will come. Then take it to this
side.
The rest of the equation will be solved.
So, if you solve this, what will be your LCM?
Here is x square, here is x and here is 1.
Your LCM will be x square. So, solve it.
Your common LCM will be x square.
Here is already x square, so 2 as it is.
Plus, if we make x square, then we have to multiply by x.
We have to do above also. 5x plus, below this is 1.
How will we make 1 into x square? We will multiply by x
square.
So, plus 3x square is equal to 0.
Now, if you take this x square to the side of 0, then it
will be multiplied.
So, it will vanish with 0.
So, what will be your equation? 3x square plus 5x plus 2 is
equal to 0.
So, which equation is this?
All of the plus plus.
3x square plus 5x plus 2 is equal to 0.
This will come, b option will be your correct option.
Now, you understood how we did this.
The reciprocal that we were given, that the roots are
reciprocal.
So, in the equation, what are roots? These are values of x.
So, we will replace x with its reciprocal.
So, our equation will come.
I hope you understood quadratic equation.
And, you will be able to ask questions easily.
Thank you for watching the video.
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