Hello everyone, welcome back to a new video.
In today's video, we are going to learn about Quadratic
Equations.
Quadratic Equations is a topic of Algebra.
First, we will understand what is an equation.
Then we will move on to what is a quadratic equation.
Before doing quadratic equation, you should know some
formulas and identities of Algebra.
I have listed them here.
You can pause the video and take a screenshot.
Let me read them and tell you what are these identities.
These are A plus B whole square.
All of you must have read this identity.
Maths, non-maths.
We have been doing this since childhood.
That is A plus B whole square.
What is the meaning of whole square?
Whole square means power 2.
For example, we have written x square.
It means any number x which is multiplied by 2.
For example, 2 square.
2 multiplied by 2.
That is 2 square.
Value would be 4.
Similarly, we have an expression.
That is A plus B whole square.
It means that we have this bracket A plus B.
How many times it is multiplying by itself? 2 times.
A plus B is a bracket.
Next bracket is A plus B.
How do we solve this bracket?
All of you must know.
First, take the first number.
That is A.
Multiply it individually by the whole bracket.
That is multiply A by A plus B.
Then, multiply B by B plus A.
Then, it will be distributed.
We have A inside.
A into A is A square.
Then, A into B is AB.
Then, B into A is BA.
Then, B into B is B square.
We have this.
Now, these two terms AB and BA are same.
We can write BA as ABB.
Now, add the like terms of both the terms.
If A is multiplying by itself twice.
If X plus X is multiplying by itself twice.
Then, it will be 2X.
Twice of that number.
Then, it will be 2AB.
This is A square plus B square.
This is the identity.
Identity of A plus B square.
That is A square plus B square plus 2AB.
Similarly, you can write minus.
In minus, you will have negative sign.
Similarly, if you have this bracket.
A plus B.
In the other bracket, you have opposite sign.
If it is plus, it is minus.
If it is minus, it is plus.
If you have two numbers multiplying with opposite sign.
Then, it will be A square minus B square.
After minus, it will be after minus.
A square minus B square.
Similarly, these are some other properties.
You will remember these.
Then, we will move ahead.
The topic that we have taken today.
That is a quadratic equation.
To know a quadratic equation.
First of all, you should know.
What is an equation?
Here, I am asking you a word.
Equation.
I am asking you another word.
Expression.
We have two sets of words.
That is equation and expression.
So, what will happen in Algebra?
In Algebra, you will have numbers.
You will have constant numbers.
1, 2, 3, half, 1 by 2.
Real numbers, rational numbers.
Any type of numbers.
You will have those.
And what will be the second thing?
You will have variables.
Variables means.
Your values which are not fixed.
For example, if I say 5.
The value of 5 is 5 for me also.
1, 2, 3, 4, 5.
And if I ask any other person.
The value of 5 will be same for him also.
But, if I give you a number.
Variable X.
So, X is only written.
There is no particular value for it.
I can say something according to me.
X is equal to 2.
You can say something according to you.
There is no value of X.
We are not taking 2.
We are taking 4.
So, the value of X is not fixed.
In every question or every situation.
What will be the value of X?
It will keep varying.
That is why it is called variable.
And the number which I have told.
5 or any other number.
It has a fixed value.
So, they are known as constants.
They are known as constants.
Constants.
So, it is variable.
Constant means any number.
It will be constant for us.
Now, moving back to the basic question.
Which I asked you in the beginning.
That is equation and expression.
Tell me about these two.
What will happen? First of all, I will start with the
expression.
If I combine variable and constant.
If I combine them.
And write something.
For example, I write 2X plus 4Y.
What will happen? How will we get 2 and 4? Constant terms.
And what will happen to X and Y?
They will become variables.
So, what did I do with their combination? I added something.
What will we call this?
What we have now.
This is just an expression.
This is an expression.
I have expressed this.
Variables and constant combinations.
Now, I can do anything with this.
I can add 2.
I can add 4.
This will be called an expression.
Now, what are the variables in this? There are two unknown
variables.
That is X.
And other unknown variable is Y.
Which we don't know.
Now, what is the multiplication of these?
That is, what is the multiplication of 2 in front of X?
This is called a coefficient.
Coefficient.
With which the constant is multiplying.
With which variable is the coefficient multiplying?
With X.
So, coefficient of X.
What will be the coefficient of X? We will get 2.
Now, the other unknown variable is Y.
So, the one which is multiplying with Y.
That is plus 4.
What will be that?
That would be coefficient of Y.
What will be this plus 4?
Coefficient of Y.
If I had written plus here.
Instead of plus, I would have written minus 4.
Then what would have been the coefficient of Y?
Then it would have been minus 4.
So, we take with sign.
If it is minus, then minus 4.
If it is plus, then plus 4.
Now, look at the last term.
Plus 2.
Plus 2 is written as single.
With that, no unknown variable is written in multiply.
And we know the value of 2.
So, what will we call this?
Constant term.
You see, this 2 and this 4 were also constant.
But, we did not call them constant term.
Why? Because they are multiplying with an unknown variable.
So, the value of unknown variable will be multiplied with
them.
So, the whole term 2X minus 4Y.
The whole term will be varying as per the values of X and Y.
So, the total expression that you have done.
Let me write it properly.
If you have written this expression.
What did you write? 2X plus 4Y plus 2. You have written
this expression.
So, the combination of these three things.
First, you have 2X.
Then, you have 4Y plus 4Y.
Then, you have positive 2.
So, the combination of these three things.
We call them as terms.
Which is separated with positive sign.
Or which is separated with negative sign.
Still, we call them as terms.
How many terms do you have for this expression? Then we
read that the variable which is not multiplied with any
other variable, we call it as constant term.
So, constant term becomes 2 plus 2, specifically plus 2.
Then we read unknown variable, i.e. those whose value is
not known.
So, how many unknown variables do you have? 2, i.e. x and y.
Then you have 2 and 4.
What will be these 2 and 4?
The coefficient of 2 which is being multiplied with.
So, 2 would be coefficient of x and 4 plus 4 would be the
coefficient of y.
We have read this much. What will we call this thing? We
will call this thing as expression.
So, this is an expression.
In how many variables? See, there are 2 unknown variables,
i.e. x and y.
Now, the second term which I had asked you, I had asked
equation.
You know what is expression.
Now, what is equation?
If we make this thing equal to any one term,
In general, we will make it equal to zero.
Whatever question you would have given, what will it be?
Make it equal to zero.
Make it equal to zero or any constant term or any term.
It can be anything, x, y, z or any number.
Then what will it be called?
If we get equal to sign, LHS comes, RHS comes, then what
will we call this?
This will be called an equation.
Now, what will this thing become?
As soon as equal to sign comes, what will this thing become
from expression?
It becomes an equation.
Now, if there is something other than zero, suppose there
is 4 here.
So, what happens to 4? You make it equal to 4. There is no
problem in that.
Make it equal to 4.
What else you can do? You can solve like terms together.
So, if you take 4 here, then it will become minus 4.
Here, zero will remain.
Then here, minus 2 will come.
So, you can bring zero here anyhow.
Whatever things are there in RHS, take them to LHS.
Reciprocate sign, means by changing it, by opposing it,
then what will you get?
Equals to, on the right hand side, zero will always come.
So, it can be non-zero also.
On the right hand side, you can always make it zero.
So, this is the equation.
And, if it is not equal to sign and RHS, then what was our
expression?
Did you understand the difference between the two?
Now, moving ahead.
In this, the variables that we had done, we had done two
unknown variables.
Now, what we will see?
We will see only the single variable thing.
That when we have only single variable instead of two.
Okay? If we have single variable, then what will be the
equation?
If it is single variable.
So, if we remove one variable.
So, keep the single variable as 2x.
2x plus 2 is equal to 4.
If I write this.
So, what is the unknown variable in this? Only one, that is
x.
So, what is this?
This is an equation in single unknown variable.
In single unknown variable.
Now, from this equation, you can find out the value of x.
Obviously, you can.
All the terms are kept aside.
And, x is kept aside separately.
So, first 2 will go this side.
So, what will happen after 2 goes this side?
Minus 2.
2x is equal to 4 minus 2.
That is 2.
Now, in multiplying 2 with this, we only need the value of
x.
So, in the next step, what will be the value of x? 2
divided by 2.
That is 1.
2 by 2.
That is 1.
So, what is the value of x? 1.
What is the meaning of the value of x?
The meaning of the value of x is.
In this equation, 2x plus 2 plus 4.
In this equation, if we put the value of x as 1.
Then, our LHS and our RHS will be equal.
And, as soon as they become equal, our equation is
satisfied.
So, this x is equal to 1.
What is this saying?
This is the solution.
Solution of this equation.
The equation which is satisfying the value of x.
That would be the solution of this equation.
So, what is x?
We have the solution of this equation.
Did you get the solution?
The value of any unknown variable which satisfies any
equation.
That would be called the solution of that equation.
Now, what is this called?
Which is in only one variable.
And, what is the power of one variable?
See the power of x.
The power of x is 1.
Now, if I am writing the equation in a single variable.
Then, I can also write x square plus 2 is equal to 0.
This can also be an equation.
What can be the next equation?
It can also be in a cube.
4x cube minus 4 is equal to 0.
It can be.
Why it cannot be?
So, any power of unknown variable of x can be.
But, what we are doing first.
First, we did this thing.
In this, what is the power of unknown variable of x?
The power of unknown variable is.
The power of unknown variable of x is equal to 1.
The power of x is 1.
So, such an equation.
See, this is an equation.
Such an equation.
In which, there is only single one variable.
And, what is the power of that? It is 1.
So, this equation is called linear equation.
What is it called?
Linear equation.
What is the linear equation?
In which, the power of unknown variable is 1.
And, what is the unknown variable?
It is 1x.
So, that will be our linear equation.
Now, I told you the second type.
That is, x square plus 2 is equal to 0.
This is our main concern.
We have to do it in this video.
That is, in such a power.
What is the power of unknown variable?
Its power is 2.
The power is written in degree.
That is 2.
What will be the name of such equations? They will be known
as quadratic equation.
Which we have to study in this video.
Quadratic equation.
Similarly, I showed you the three power equation.
So, three power equation is also known as cubic equation.
And, so on and so forth.
There are some specific names of such powers.
But, now what is our concern in this video?
That is your quadratic equation.
In which, what will be the power of unknown variable?
Its power is 2.
And, what is the unknown variable?
Its power is 1.
That is, x.
So, if I write the general equation of quadratic equation.
You see, this was the question which I wrote in the last.
x square plus 2 is equal to 0.
In this, the power of unknown variable was 2.
Now, you have to pay attention to one more thing.
That the highest power of unknown variable should be 2.
Highest power.
So, what is the change in the definition? That the highest
power of unknown variable should be 2.
Otherwise, x cube would have come here.
Now, would this be a quadratic equation? Because, here x
square is there.
So, is this a quadratic equation?
No.
Because, here one more power has come.
That is, x power 3.
That is cubic equation.
So, when the highest power of variable becomes 2.
Then, it will be called quadratic equation.
Okay? So, what we will write?
I will write the general form of quadratic equation.
General form.
General form of quadratic equation.
general form of a quadratic equation, what will happen to
it? a x square should be of highest 2 power only, then
lowest degree can be any degree below this, but at most
degree should be 2, which is 2 and any degree can be
smaller than that.
So, first take 2 degree x, because it should be of highest
degree, then after 2, it will be of 1 degree, so x multiply
coefficient of b, then of 0 degree, so whenever x power is
0 or any term's power becomes 0 of any number, then it is
equal to 1.
So, x power 0 will be 1 only, with that any constant can be
multiplied with it, so write constant with c.
So, this is general expression for quadratic, now we have
to make equation, this is now quadratic expression, now how
to make equation, write this as equals to 0, so this is
your equation, this is your general form of a quadratic
equation, we can write
any quadratic equation in this form, where a is, what is
coefficient of a? Coefficient of x square, b coefficient of
y square, sorry
and c, c is your constant term. So, this is our general
form of a quadratic equation.
Now, when we did linear equation, then we got the answer of
this expression, of this equation, we got the answer, what
we got? x is equal to 1, we got the answer, x equals to 1,
solution, I said this is your solution, means this value of
x will satisfy
this equation.
So, can we get the solution of quadratic equation also?
Obviously, if we are starting to do any equation or
anything in maths, then it will surely get some answer,
right?
If we are solving any question, then answer will come, if
it will not come, then means researchers have not found
out, we have not reached that level, we should get the
answers of everyone, so we will see that what will be the
solution of this quadratic
equation?
Now, what do I mean by solution? From solution, that
unknown variable, what value of variable
do we put? What is the variable? x is unknown variable, so
what value of unknown variable x do we put, so that LHS and
RHS will be equal.
So, what is the formula of its solution?
You can write the formula of its solution, slowly when you
will increase in maths, you will also learn how to derive
that answer.
But as of now, we do not go towards the proof, we write the
answer directly, that is, what values of x will come? Now,
the last equation which we did, linear equation, so in that
we got the particular value of x, it was a linear
equation. Now, in this, x power is 2, so you see, whenever
you used to do square, whenever you have x square is equal
to 4, so you will put under root, so from under root, you
will get two answers like this, one is plus 2 and the other
will be minus 2,
so you will always get two answers of square power, so
same, how many answers will we get of quadratic equation?
Two answers, how many values of x will come? Two values
will come, what will be the values? Write the formula,
minus b, minus b plus minus,
plus minus under root b square minus 4ac whole divided by
2a, whole divided by 2a.
See, plus minus, means once we will take plus, once
negative term, so we will get the answer, two values of x
will come.
So, what will happen? Minus b plus minus under root b
square minus 4ac divided by 2a, a is coefficient of x
square, b is coefficient of x and c is constant term.
So, what will happen? These are two solutions of this
quadratic equation, two solutions.
Either you say solutions or you say, other than solution,
the other term is root, root.
Why we say root? Because whatever value of x we will put
here, what should be the whole side of LHS? Whole LHS
should be zero, because only then it will be equal to LHS.
That is why we say it as root of the equation.
Now, just to analyze a little, I will tell you its graph.
What is its graph? Its general graph is like this, U.
Now, how can that U be? U can be passing through this gap,
it can be passing through below zero, it can be passing
through above zero, it can be passing through right shift,
it can be passing through left shift.
Why this gap will be there? There are further cases of this
in big classes, means in advanced topics.
But here if you have to give some attention, then you
should just pay attention to this.
What will be its graph? Its graph will be U.
Now, how will it be? We will see that in the next questions.
If you have questions of this type, then we will definitely
see.
So, what is this? Your solution. Remember all the solutions
once again. Read it quickly.
Remember.
Now, what do we do with this? Let us write it again.
What is the value of x in the formula?
Now, one thing which is very important in this, we will
tell you.
Till now, you have taken out the root and its values.
Will you be able to take it out if A and B are given to
you? Obviously, you will be able to take it out.
Now, we will know nature.
What is the nature of those roots? Nature of the roots.
Now, what is the nature of the roots?
Nature of the roots.
Nature as in, what type of solutions are we getting? Are we
getting real numbers or are we getting imaginary numbers?
Are they equal or unequal? What can be their
nature? We will know that from this formula.
Now, see, the main thing which we need to know is, we need
this inner thing.
Because, this minus B will remain as it is.
Whatever will be the value of B, you keep it.
Keep 2A.
Whatever will give difference in our solution,
we will know that from this term.
From which term? That is, from B square minus 4AC, we will
know the nature of solution or the nature of roots.
So, write B square minus 4AC.
denote this with capital D. Why with D? Because, its name
is discriminant. We call it discriminant.
Now, B square minus 4AC.
Whatever we solve, we will get a number.
Now, if you see the real line, then how many parts are
there in the real line? Real line is 0 in the middle, then
positive numbers on the right hand side, negative numbers
on the left hand side.
So, if we see the total, there are 3 categories.
One is of 0, then positive numbers and then negative
numbers.
So, what can be their values? It can be 0 or it can be
positive or negative.
There are 3 possibilities.
So, the first possibility is that if we have D, then we
have greater than 0.
What is D? We have greater than 0.
That is, this whole term B square minus 4AC, what we have?
Positive.
This whole term is positive.
What will we get? Positive.
If you are positive, then positive in the root. What will
be the value? It will be a rational number or it can be
irrational also.
But basically it will be a real number.
If you divide 2a by any other number, what will it give?
It will give you a real number.
Then your roots will be positive.
First of all, if it is positive, then it will be a real
root.
It will be a real solution.
And what will be the answer?
It will be 2. One will open in plus and one will open in
minus.
So will these two values be equal?
Suppose its value is root 4.
Then what will happen?
So once it is open, what will happen?
What will you do to minus b plus root 4?
Once it is this value and the second time it is minus b
minus 2 divided by 2a.
If both are unequal, then what will be the values?
When it is positive, then it will be a real solution.
First of all, the determinant will be real.
The solution will also be real.
And what will be the solution?
These two will be unequal.
Both the roots will be unequal.
Any sign can be positive or negative.
We will know that b is less than this or less than 2.
We will know the relation between b and this.
Then we will know whether it is positive or negative.
But if we see value wise, then both will be unequal.
So when will this case happen?
When your d is greater than 0.
Let us see the second case.
In the second case, if d is greater than 0.
I am erasing this.
If you have written the second case, then you can write it
in the back.
You will get the second case if d is equal to 0.
d is equal to 0.
This means that the whole term is equal to 0.
b square minus 4ac.
So the under root of 0 will be 0.
So what is the solution left?
The whole term will be 0.
So what is left?
We will have this.
This would be x is equal to minus b 0 divided by 2a.
Did you get the answer?
So what is left?
We will get the real solution.
And what is the number like this? We will get only one
solution.
When did we get 2?
When we had this plus and minus sign.
In this case, we will get only one value.
Whether you keep b positive or a negative.
Whatever is their given value.
The answer that will come, that would be only one answer.
You will not get two values of x.
Because the portion giving two values is not there.
It becomes 0.
So the answer will be the same.
In this case, the roots will be real.
How will the roots be?
They will be equal.
In that case, the roots were real but unequal.
In this case, the roots are real and equal.
Both are real and equal.
So this was the case when d is equal to 0.
So it is greater than 0.
Now the third case.
What will happen in the third case? When it is less than 0.
For less than 0, write the third case.
When d is less than 0.
When d is less than 0,
It means that we have a negative answer in the root.
Till now, did we know that there is a negative answer in
the under root?
We did not know this.
We were always taking positive values in the under root.
But now we have a negative value.
For example, the root of minus 5 is negative.
So how will we solve this?
Whenever there is a minus sign in the under root,
Then we do not have the real term.
Now when we have this term,
This would be our imaginary number.
The number of imagination will come.
That is imaginary number.
And the remaining terms will remain as it is.
Minus b and divide by 2a.
So when we have d less than 0,
Then we will have unequal roots.
And they will not be real.
This time they will be imaginary.
That is, this case will be solved in complex numbers.
Till now, we were doing all real cases.
But now we have a complex case.
So you have understood the nature of roots.
You have seen the discriminant through the solution.
You have seen the three cases of discriminant.
And you have seen the nature of the roots.
Now the roots that come,
What type and how will we find them?
We will see that.
What will come and how will we find them?
Write the same general equation.
The roots that come in this will have their value.
We will do the basic question of this.
Will the roots come out?
We will see the equation.
You can see any equation.
We have this equation.
First of all, compare it with this.
Find out the value of a.
The value of a is 2.
What is the value of b? 4 And what is the value of c?
Minus 3
You will not write the value of c as 3.
You will write it with minus sign.
That is minus 3.
Now put the value in it.
x is equal to minus b.
That is minus 4.
Plus minus under root.
b square that is 16.
4 square minus 4 into a.
That is 2 into 3.
That is minus 3.
Value of c.
Divided by 2 into a.
That is 2.
Solve this.
4 x 3 is 12.
12 x 2 is 24.
Minus minus plus 24.
That is minus 4 plus minus under root.
16 plus 24.
Divided by 2 into 2 is 4.
16 plus 0 is 1.
40 root 40 divided by 4.
If you solve this further, what will be this?
Minus 4 plus minus.
How can you write this?
Root 4 into 10.
Divided by 4.
How can you get root 4? I am solving this term.
This would be root 4 into root 10.
Value of root 4 is 2.
2 will come.
Solve this.
2 into root 10.
2 root 10.
This will be cancelled if we take 2 common.
If we take 2 common, this will be cut from 2.
This will be cut from 2 and this from 1.
This will be minus 2 plus minus root 10.
Divided by 2.
What is your solution?
There are two solutions.
Minus 2 plus root 10.
Divided by 2.
Minus 2 minus root 10.
Divided by 2.
This is your second solution.
You can put values and find solutions.
You have put this in formula and found it.
But remember the formula.
That is the main point. formula is the same as the sum of
two roots.
Minus of sum of roots.
Minus of minus of sum of roots.
Minus of sum of roots.
Minus of minus of sum of roots.
Minus of sum of roots.
Minus of minus of sum of roots.
Minus of sum of roots.