In the chapter on Quantic Equation, we will study the theory of equation and many other things related to equation. You have to study this chapter very carefully because if you study any chapter in Maths, you will find some work in it. We will start the quadratic equation with quadratic expression. We will start with quadratic expression.
Please understand this. Then we will come to the quadratic equation and its properties, use and questions. So, we will come to the next question.
Qualitative expression. Any expression, ax2 plus bx plus, what is this called? Quadratic expression. Why is it called a quadratic expression?
Because the highest degree is 2. So, this thing is called a quadratic expression. Is this clear? This quadratic expression is giving us a relation between two variables.
Which two variables? x and y. Since you are getting a relation between two variables, we can easily make this expression.
Graph. What will be the graph of a quadratic expression? Parabola. If there is a quadratic relation between any of the two variables, y equal to ax square plus bx plus c, then if we plot y versus x, then what will we get as a graph?
Parabola. Now listen, parabola will either be like this or like this. We will get two parabolas.
Either the parabola's face will open upwards like this or the parabola's face will open downwards like this. This point is called vertex. The legs of parabola are going to infinity.
Where are they going? They are going to plus infinity. And where are these legs going?
They are going to minus infinity. If the top of parabola is opening, So this vertex will be its lowest point. The smallest point. And how far will these legs go? Plus infinity.
And if the parabola is opening downwards, then how far will these legs go? minus infinity and in case of vertex, it will be highest point. Now the question arises that the graph of quadratic expression is parabola, but how will we know whether the graph will open up or down? How will we know? By looking at the sign of coefficient of x square.
What is the coefficient of x square? a. Either a will be positive or A will be negative if A is positive then where will the mouth of parabola open? and if a is negative then where will the mouth of parabola open?
Down. Did you understand? Keep in mind that if the mouth of parabola is opening upwards, then where are the legs going?
Plus infinity. And if the mouth of parabola is opening downwards, then where are the legs going? Minus infinity. Did you understand?
Now this mouth of parabola is opening upwards, it may be on the x-axis, it may be touching the x-axis, it may be If the parabola is facing downwards, then it may be below x-axis. It may be touching x-axis. It may be cutting x-axis. How will we know if it is touching, cutting or intersecting x-axis? By discriminant.
What is the special thing associated with this quadratic? Discriminant. If we say discriminant, b square minus 4ac. If the discriminant is less than 0, then the parabola will be away from the x-axis. That is, either it will be flying over the x-axis or it will be below the x-axis.
If the discriminant is equal to 0, then the parabola will touch the x-axis. If the discriminant is equal to 0, then parabola will touch x-axis and if discriminant is greater than 0 then parabola will cut x-axis in two places and if discriminant is greater than 0 then parabola will cut x-axis in two places First of all, understand quadratic expression because this is the most useful How will the graph of quadratic expression be made? Parabola Either the upper part of parabola will open or How do we know if the parabola is up or down? By giving the sign of x2. If a is positive, then the parabola is up.
If a is negative, then the parabola is down. Is it clear? After opening the mouth, we have to see whether the parabolic is touching the XX or cutting it, touching it or not.
So how will we know? By looking at the discriminant. Is it clear? Is there any problem in this? Is it clear?
Okay, now listen. The lowest point is that no matter what the graph is, the x coordinate of the vertex is always. No matter what the graph is, the x coordinate of the vertex is always.
Tell me one thing, if we know the x coordinate of vertex, then can we find the y coordinate of vertex? How? This is the relation.
This gives us the relation between x and y. So if we know the x coordinate, then we will get the y coordinate. If you keep the value of x as minus b by 2, then the y coordinate will be minus d by 4. Where will it come out? It is minus d by 4. Here also, the y coordinate is minus b by 2. minus d by 4a.
See, if d is 0, then y coordinate will be 0. If d is 0, then y coordinate will be 0. Here also y coordinate minus d by 4a. Here also y coordinate minus d by 4a. Here also y coordinate minus d by 4a. And here also y coordinate minus d by 4a. 4a means any graph which will have vertex x coordinate minus b by 2a and y coordinate minus d by 4a If we have any Quartic expression, whether the face is facing up or down, whether the face touches the axis, cuts it, intersects it, stays away from it, or does anything, the x coordinate of the vertex will always be 2a.
And the y coordinate of the vertex will always be 4a. We have to keep this in mind. In any case, the coordinates of the vertex will never change. Do you understand? One more thing, where are these legs going?
plus infinity and here the tang is going to minus infinity. Did you understand the graph? Is there any problem in the graph? Is the graph clear?
Ok, one more thing. If mu is opening upwards, pay attention, if mu is opening upwards, if mu is opening upwards, then what will be the minimum value of y? minus d by 4. And what will be the maximum value of y?
infinity. Here, if mu is opening downwards, is opening, then in this situation, what will be the minimum value of y? Minus infinity.
And what will be the maximum value? Minus d by 4. What do you understand? You can see where the legs are going.
It is minus 1. To write separately, I will just write that if a is positive, then the minimum value of y will be minus d by 4a. Say yes. And from here, if a is less than 0, then the maximum value of y will be minus d by 4a. Did you understand the graph? Any problem?
Is it clear? Okay? Did you understand?
Now see, one moment please. There are other things related to this. Now listen, the graph that we just saw, there are other things related to this graph. Please pay attention.
We just talked about quadratic expression. We are discussing about y equal to ax square plus bx plus c. We were talking about this.
This is quadratic expression. What will be the graph of this? parabola.
What will be its graph? parabola. We have already seen that. What is the vertex of parabola?
All these things are clear. Now we will discuss many more things from this graph. We will discuss them one by one.
First of all, I want to say that when will this happen? That AX square plus BX plus C is greater than 0 for all X epsilon R. When will it happen that, no matter what value you keep for X, but AX square plus BX plus C, this quadratic expression will always be greater than 0. When will it happen? Think. If you want it to always be greater than 0, then which graph can work out of these?
Will this graph work? No. Why?
Because here x is negative. Will this graph work? No.
Why? Because 0 is coming on this x. But we don't have 0. Will this graph work? Yes.
So this is the only graph. That means if you want a greater than 0 for every x value. If you want a greater than 0 for every x value. So what should happen? A greater than 0 and discriminant less than 0. If a is greater than 0 and discriminant less than 0, then you can keep any value of x.
This quartic expression will always be greater than 0. Is it clear? Okay, wait a minute here. If I ask you here that ax square plus bx plus c is greater than or equal to 0 for all x epsilon.
Which graph? No, this one will also work. Both these graphs will work because in this only 0 will be less than or equal to 0. So the condition will be a greater than 0 and discriminant less than or equal to 0. and the discriminant will be equal to 0 and less than 0. Is it clear? Now if I ask you that Ax square plus Bx plus C is less than 0 for all x absolute.
Keep any value less than 0. Which graph? Can this graph work? No.
So, we have to keep less than 0. That means, a less than 0 and discriminant b less than 0. Is it clear? And here, if I ask you, a x square plus b x plus c is less than or equal to 0 for all x epsilon r. Keep any x square, always less than or equal to 0. So, this graph will also work.
and this will also be different if it is less then it is ok and if it is equal to 0 then it is ok this time we will put condition a less than 0 and discriminant less than or equal to 0 catch the point when quadratic equation will always be positive when quadratic equation will always be greater than 0 when quadratic Less than zero and when the quadratic is less than or equal to zero? Did you understand this thing related to the graph? Is this clear?
Very good. Look, we are talking about quadratic expression. Quartic expression of y is equal to ax square plus bx plus c. This is clear. We have seen the grass.
We have seen when the quartic expression will always be positive. When the quartic expression will always be negative. And so on. Come, let's solve some good questions.
And pay attention to the illustrations. So that you can understand the whole thing. Clear? Now listen carefully. A small question says, if Ax square plus Bx plus C equal to 0 has no real roots.
Listen carefully. If Ax square plus Bx plus C equal to 0 has no real roots, then show that C into a plus b plus c divided by 0. Listen carefully. The question is, if there is no real root of a x square plus b x plus c, As soon as we hear in our ears that there is no real root of a x square plus b x plus c, what does this mean?
Look, in this quotation, How many graphs can there be? 6 graphs? How can there be 6 graphs? 1, 2, 3, 4, 5, 6. Either the mouth opens upwards and the air keeps turning. Or the mouth opens upwards and touches.
Or the mouth opens upwards and cuts. Or the mouth opens downwards and stays away. Or the mouth opens downwards and touches.
Or the mouth opens downwards and cuts. Did you understand? Now we want that there is not a single real root. There is no real root. No real root.
This means that this possibility cannot be there. This possibility cannot be there either. Real root means that at any value of x, this 0 will not go.
If at any value of x, this 0 will go, So, for every x value, it will be either positive or negative. What do you think? If you ever heard that there is no real root of this quadratic, it means that neither the real root of this quadratic, x axis is not touched nor cut by x axis.
Real root is not there. That means, no matter what value of x you keep, this thing will never go to zero. No matter what value of x you keep, this thing will never go to zero.
As soon as no real root is there, Listen to the real root. Either the graph will be like this or the graph will be like this. Do you understand this inference?
We should never listen to the real root. This means either every x value will always have a quadratic positive or every x value will always have a quadratic negative. This means at all values of x.
Either ax square plus bx plus c is positive or negative. This will always be positive on all the values. Or it will always be negative on all the values.
It can never be zero. Is this clear? This is very important.
This is used 2-3 times in IT. Okay, so if I keep any value of x, either this will always be positive, positive, positive, or this will always be negative, negative, negative. Do you agree with this?
If for a short time, Ax square plus 2 bx plus c, we call fx. So I can say that either fx is always positive or fx is always negative. This or this. Do you agree with this? Do you agree with this?
Now see what I am saying. C into a plus b plus c. If you say this to fx, then clearly you guys see that the value of f of 1 is a plus b plus c. We will keep 1 here. And see the value of f of 0 is c.
And either both of them will be positive. Or both of them will be negative. Why?
Because if the graph is like this, then any value of x will always be positive. And if the graph is like this, then any value of x will always be negative. Can I say that both have the same sign? Either both of them will be positive or either both of them will be negative.
Is this clear? Is this clear? If both have the same sign, then the product of both will be F1 into F of 0. That is, A plus B plus C into C. What will this always be?
Positive. If both have the same sign, then both will be positive or both will be negative. In any case, the product will be positive. Is the question clear?
Is the question clear? I am going to note two things on which I am going to develop the whole chapter. It is never said that it has no root.
It has no root, which means it does not go to any x value. It does not go to any x value. then neither will it touch nor will it cut the x axis. Because if it touches the x axis, then 0 is going for x. And if it cuts the x axis, then 0 is going for these two x's.
This means that the graph is always up and the graph is always down. That is, for every value of x, either f will be positive or for every value of x, f will always be negative. Is this clear?
Copy it. Now we have discussed about no real root. Let's see one more thing on that no real root.
Question says if ax2 plus bx plus 1 equal to 0 has no real root then show that a plus b plus 1 is greater than 0. Listen carefully, very carefully. There is no real root of aX2 plus bX plus 1. Now since there is no real root of this, we understood that either its graph is floating in the air, or its graph is in the ground. is the same as the value of x.
If the graph is floating in the air, then the value of x will always be positive. And if the graph is in the ground, then the value of x will always be negative. Let's look at this expression. This is the expression ax2 plus bx plus 1. What I am saying is that there is no real root of this.
Either it will be positive on every x value or it will be negative on every x value. Do you agree with this? Is there any problem?
Let's take f of 0. So, the value of f of 0 is 1. I thought of f of 0 because it has given the value of a constant term. What is f of 0? Positive.
If the graph is this, then every x value should be positive. And if the graph is this, then every x value should be negative. And if we leave these two graphs and go to the third graph, then it will not happen.
It didn't happen. But if this graph was there, then the negative value of Rx should have been there. But F0 is positive.
This graph rejects and this graph accepts. We can say from all these things that this expression has a similar graph. Did you understand?
There were six graphs in total. We rejected four graphs saying that it has no real root. And the value of constant term was kept here.
The constant term was positive. If f0 is positive, then this graph is not possible. Because if you take this graph, then what should be the value of every x?
Negative. But what is the value of 0? Positive. So the graph will be only this. Do you agree with this?
Do you have any problem with this? Now think again. What value of x should I keep that becomes 1? How much will be f?
a plus b plus 1 Now this is obvious, this is the graph So what will be f? Positive So what will be f? Did you understand this? As we have seen no real root, no real root means either the value of every x is positive or negative for every x. Is it clear?
Is it clear? Is there any problem? Look at one more question. A very good question came in IIT. Come, let's do the question in that very good IIT together and see whether it is made of us or not.
Listen carefully. The question says, Question says, find value of m so that y equal to x square plus 2m plus 6x plus 4m plus 12 is always positive for all x epsilon r. The question is, find the value of n so that it is always positive for all xs.
Now tell me, help me, will this quadratic expression always be positive? The quadratic expression will be positive only when the quadratic equation is positive. the a positive one discriminant negative other a positive discriminant negative yoga the quality expression of a shaggy over positive I'm chatting up X key code available yeah I'm a shop positive I as a time will happen when its graph is floating in the air. And the graph is floating in the air, the mouth is open, so a greater than 0 and discriminant less than 0. Any problem?
What to solve in a greater than 0? The coefficient of x square is 1. So 1 greater than 0 is always 0. Say yes. Say yes. Discriminant less than 0. Solve it. What is the discriminant?
b square minus 4ac. This should be less than 0. Am I right? Solve it.
If we do the squaring here, 4ac. It is correct. b square minus 4. Let's do a simplification. If we take 2 out, square will be minus 4 into 4 m plus 12 will be less than 0. Divide by 4, then we get whole square of m plus 3 minus 4 m plus 12. M square plus 9 plus 6M minus 4M minus 12 less than 0. So, this thing is M square plus 2M minus 3 less than 0. Are there any factors? Are there any factors?
m plus 3 and m minus 1. Am I right? We know the time-at-all-interval. We will make it number 9. Where will m plus 3 start? Minus 3 plus 1. So positive, negative, positive. We want negative.
So, it is clear. The solution has come out. Where did the value of m come from?
Minus 3 to the power of 7. So, if you keep the value of m minus 3 to the power of 7, then what will always come out? Negative. What will always come out?
Negative. Sorry. Always positive will come out.
X for epsilon r. Keep any value of x. The value of the expression will always come out to be positive. Is it clear?
Is it clear? Is it clear? Now let's do one more question like this. Look at one more question.
One more question. One more question is Find A so that expression A plus 4x square minus 2a x plus 2a minus 6 is less than 0 for all x epsilon r. Let's talk about it.
Listen. We have to find a value like this. so that the expression is always negative. Listen to me carefully. If this expression has to be always negative, then how should the graph be?
It should be like this. So that what is always on the value of every x? Negative. That is, the face should be open downwards and stay away from the x-axis. So when will the expression be negative?
When the coefficient of x square is less than 0. And discriminant is also less than 0. So the condition will be this. Coefficient of x square. That is, a plus 4 is less than 0. Say yes. And discriminant.
That is, b square minus 4ac. This will also be less than 0. So from here a is less than. minus 4. Is it clear? Let's divide this by 4. If we divide by 4, then a2 minus a plus 4 into 2a minus 6 less than 0. So, here a2 minus, let's multiply this. 2a into a, 2a square.
2a into 4, 8a. Then minus 6a minus 24 less than 0. Am I right? So, a2 minus 3a. 2 a square, 8 a minus 6 a 2 a, 2 a over minus sign minus 2 a plus 24 less than 0. So a square minus 2 a square minus a square minus 2 a plus 24 less than 0. Minus sign multiply it over, so a square plus 2 a minus 24 greater than 0. 24 or 2, can it be a factor?
6 4 is equal to 24. So here we have a plus 6 into a minus 4 is equal to 0. So if we plot the number line, what will be the value of this? Minus 6. What will be the value of this? 4. And positive and negative?
So we need a greater than 0 or greater than 4 or smaller than minus 6. But we also have to look at this. We have to take the intersection of both. We have to take the common of both.
So it is saying that it is smaller than minus 4. So minus 4 will come here somewhere. So, the common of both will be this part. So, the net answer will be minus 6 from infinity.
Is it clear? Look at another question. There is another very good question which is similar to this one. It was a simple question.
But, let us do it together. The question says that, Find value of k so that k minus 2 x square plus 8 x plus k plus 4 greater than 0 for all x epsilon r. Now we have to find the value of k so that this expression is always positive.
Now tell me when this expression will always be positive? When its graph becomes something like this. That means what is always found for every value of x?
Positive. When will this be? When xA0 and discriminant 0. A0 means x2 is positive.
So what condition will be? k-20 means k2. Any problem? Discriminant 0. How much will be discriminant?
b2. So 64-4ac. What should be? Less than 0. Say yes.
Divide by 4. So, 16 minus k minus 2 k plus 4. Let's say yes. So, 16 minus k square. Here, it will be plus 4 k minus 2 k plus 2 k minus 8 less than 0. Open the bracket. 16 minus k square minus 2 k plus 8 less than 0. So, here it comes. Minus k square minus 16 and 8. 24. Multiply by minus sign.
Plus 2k minus 24 is 1.5 times 0. Wow! What will be the factorization? k plus 6 into minus 4. Am I right? Plot the number line. This will be minus 6 and 4. Positive, negative, positive.
Either this is greater than 4. or minus 6 is smaller than 2 but this is bigger than 2 so we have to take the common of both so this is bigger than 2 so net common is coming out here so finally k is 4 to infinity Listen carefully. This is a quadratic expression. Ax square plus bx plus c is a quadratic expression.
Now we will pay attention to this. We will see the graph of ABC. Just by looking at the graph, you can see the graph of ABC. It is possible that the graph is like this It is possible that the graph is like this I have made some graphs In each of these graphs, we have to tell the sign of ABC It is very easy to tell the sign of A by looking at the face What will be A in this graph? Positive What will be A in this graph?
Negative What will be A in this graph? Negative What will be A in this graph? What will be A in this graph? Negative. What will be A in this graph?
Negative. Any problem? Did you find out the sign of A?
Now tell me how will you find out the sign of C? The sign of C. Now listen, if this is fx, ax square plus bx plus c, then f0 equal to c, So, I can say that at x equal to 0, the y coordinate is c.
Am I right? Where does the graph cut the y axis? 0, C.
Any problem? Can I write this? Can I say this?
That graph cuts y-axis at 0, C. Can I say this? Can I say this? Now look here.
Graph is cutting at 0, C. So it is cutting at positive y. So what will be C here?
Positive. Say yes. Here also the graph is cutting at 0, c. So c will be positive. Here also the graph is cutting at 0, c.
So c will be positive. Because x axis is above and y coordinate is positive. Here also graph is cutting at 0, c so c will be positive.
But here graph is cutting at 0, c so c is clearly negative. Here graph is cutting at 0, c so c is clearly negative. To tell the sign of C, you will have to see where the graph Y axis is cutting.
If the graph Y axis is cut in a positive place, then the value of C will be positive. And if the graph Y axis is cut in a negative place, then the value of C will be negative. Did you understand? Now tell me, A sign is cut by looking at the face, C sign is cut by looking at the graph.
How will you see the sign of B? How will we see the sign of B? I will tell you. Vertex. What is the x coordinate of vertex?
Minus b by 2. Come here. Here vertex is positive. Think. That means minus b by 2 is positive.
A itself is positive. So, minus should be positive. Minus should also be positive.
Whenever it is negative. Do you understand? I am writing a summary here.
Look at it. How will you see the sign of A? By looking at its face.
Whether its face is up or down. How will you see the sign of C? Where? Cuts y-axis.
Where does y-axis cut? And how will you see the sign of B? Vertex. From the location of the vertex.
Did you understand this? If you look here carefully, So, minus b by 2a is positive. Vertex's x coordinate is positive.
a itself is negative. So, negative upon negative is positive. So, b should also be positive. Did you understand? Did you understand?
Here, oh! Here, minus b by 2a is negative. a is also negative. So, negative upon negative is positive. So, what should be B?
Negative. Are you getting it or not? Here, see, here minus B by 2A is negative.
Vertex's x coordinate is negative. A is positive. So, this is positive. So, minus B has to be negative.
Minus B has to be negative. So, B has to be positive. Did you understand? Here, see, vertex's x coordinate minus B by 2A is positive. a is negative so a is negative and this is negative so it is positive so b should be positive here see minus b by 2 is negative a is negative so negative over negative is positive so b should be negative So, Basant brother, you can tell the sign of A, B, C by using the graph.
Where is the A sign opening? Up or down? C sign will tell where the y axis is cutting. And where will B sign tell?
By looking at the location of the vertex. Is minus b by 2 a positive or minus b by 2 a negative? Quality difference.
Now I am going to tell you something amazing. Listen to this. Look, the previous study we did, if a is greater than 0, then all the children will believe that the graph will open up. So the minimum value of y will be minus d by 4. If a is less than 0, then the graph will open up.
So the maximum value of y will be minus d by 4a. This is the minimum value when you are taking the whole x epsilon r. This is the maximum value when you are looking at the whole graph.
What does it mean to look at the whole graph? Listen, let's assume this is our quality graph. Now if I ask you to tell the minimum value of this quadratic, then the minimum value will be here.
But if I give you a small interval, that tell the minimum value in this interval, then the minimum value will be in this interval. And this is the maximum value of the interval. Let's start again.
In this case, the vertex value is not equal to the minimum value. If I separate the interval, suppose I say, tell me the minimum value of this interval. This interval is at the vertex value of the minimum value.
And the maximum value is at the vertex value. Clear? Now I will discuss about Minimum and Maximum value in an interval.
We were taking out its minimum in x epsilon r. We were taking out its maximum in x epsilon r. But we took a small interval. In this interval, tell me where is the minimum and maximum? In this interval, tell me where is the minimum and maximum?
To explain this, we will take the question in IIT and we will also understand the question and how to see minimax in the interval. There was a very good question in IIT. There was a question in IIT that find the value of p so that quadratic this 4x square. minus 4px plus p square minus 2p plus 2 has minimum value 3 in x epsilon 0 to 2. Question is great.
This question is great. So, you have to pay attention to this. It is being said that take out such a value of P so that this expression has minimum value in this interval.
value 3. This expression's minimum value in this interval is 3. Is this clear? Is the question clear to everybody? This expression's minimum value in this interval is 3. Is it clear?
Let's see. If the interval is 0 to 2, then how will we know the minimum value of this interval? How will we know the minimum value of any interval? I will tell you how to know.
Look at the parabola graph. The parabola graph first decreases and then increases. I would like to say a small thing.
If the entire interval is before the vertex, then the minimum value will be at the end point of the interval. Because the graph before the vertex is decreasing. If the entire interval is after the vertex, then the graph at this interval is decreasing.
increasing the minimum value will come at the beginning and if the vertex is inside the interval then the minimum value will be lost on the vertex. Listen again. The graph of parabola first decreases then increases. So where will the minimum value come in the interval?
If the entire interval comes before the vertex then minimum at the end of the interval. The entire interval comes after the vertex. Minimum at the beginning of the interval and the vertex at the end of the interval. Then the minimum is clear at the vertex.
So I have to find out where the minimum will be at this interval. So this will depend on whether this interval is before the vertex Is this interval after the vertex or between the vertex intervals? For this, we will first find the vertex of this interval.
Where does the vertex come? It comes as minus b by 2a. So where is the vertex?
It is b by 2. Tell me. This is the x coordinate of the vertex. Now, there can be three things. Either the interval is done before the vertex.
If the interval is given before the vertex, then p by 2 is greater than 2. Because in this situation, the interval will be given before the vertex. In this situation, p should be greater than 4. In this situation, the minimum value will be 2. So in this situation, f of 2 should be equal to 3. This is what it says, that the minimum value of this interval should be 3. This is one case, when the entire interval falls before the vertex. The second case, that the interval falls somewhere in the vertex interval.
So it will fall somewhere in the vertex interval if p by 2 is between 0 and 2. That is, p. It will be between 0 and 4. So this vertex will be in the interval. So where will the minimum value be?
It will be at the vertex. In this situation, f of p by 2 will be 3. But the minimum value should be 3. There will be one more case. There will be one more case that the interval will be after the vertex. How will it be after interval vertex? That 0 to 2 will be after interval vertex.
It will be after PY2 less than 0. That is P less than 0. In this situation, minimum value will be 0. So, F of 0 should be 3. Actually the problem is that he is asking us to tell such a value so that the minimum value in this interval is 3. Minimum value in this interval is 3i. So we don't understand where the minimum value will come. So where will the minimum value come? Will it come at 0?
Will it come at 2? Or will it come somewhere else? First of all you have to see where the vertex is. Because if it is after the vertex interval, then the minimum value will come at the end.
If the vertex is before the interval, then the minimum value will be in the beginning. And if the vertex is not in the interval, then the minimum value will be on the vertex itself. Did you understand? Let's solve this problem. Will there be any problem if there are three cases?
F of 2 should be 3. Keep 2 here. 2 square 4. 4 4 is 16. Keep 2 here. 4 2 is 8. Here, plus p square minus 2p plus 2. This should be 3. Come, p square minus 8p minus 2p minus 10p. 16, 17, 18 plus 15 equal to 0. So, p's value is minus b plus 2p. minus root of b square 100 minus 4ac 15 4 is 60 so 100 minus 60 is 40 upon 2 so p value is 5 plus minus and therefore 2 root 10. So, P is equal to 5 plus root 10 or P is equal to 5 minus root 10. But P4 should be greater than this.
So, we reject and accept this. Because when did this case come? When minimum value will be 2 when P4 is greater than this.
So, we will accept the same solution. which is greater than 4. This is the condition. This is true only when this condition is present.
I can write this. If this, then this. Did you understand?
So one value is left, 5 plus 1. Any problem? Look here. F of p by 2 is 3. So keep p by 2 here.
So p square by 4 is p square. Keep p by 2 here. So minus 2p square is left.
So, P is equal to minus half of minus 2P. Reject. So, P will be accepted as the same value as 0 to 4. So, when did this whole case come into being?
When P is between 0 and 4, do you understand? Do you understand? Do you understand? Next, do F of 0 is equal to 3. So, both are 0. So, P square minus 2P plus 2. This should be 3. So, P square minus 2P minus 1 should be 0. Now, solve this.
Minus B plus minus root of B square. 4 minus 4 is 8 upon 2. So, P value is 1 plus minus root 2. So, P value is 1 plus root 2 and 1 minus root 2. But p should be less than 0, so we reject and accept. Keeping all the things in mind, we have reached this conclusion. p should be either 5 plus root 10 or p should be 1 minus root 2. This is the final answer.
Did you understand? Wait a minute, wait a minute, wait a minute. Did you understand or not?
Did you understand? Listen to the last thing I want to tell you. If we have to see max or min value in an interval, If we want to see the minimum x value in interval, then first of all we will see the position of vertex with respect to interval. Position of vertex, then see position of vertex with respect to interval. If the interval is before the vertex, then minimum is at the end and maximum is at the beginning.
If the interval is after the vertex, then minimum is at the beginning and maximum is at the end. If the vertex is in the interval, then vertex is at the minimum and maximum is at the end. If you want to see min and max in the interval, then you will first see the location of the vertex.
Only then you will know where the mean and where the minus will come. Thanks. Copyright 2020 Mooji Media Ltd. All Rights Reserved. No part of this recording may be reproduced