 Hi friends, so in the previous session we discussed about the sign of a quadratic expression given by ax square plus bx plus c and what is its relation with the sign of this expression and its relation with the sign of the coefficient of x square that is a right and we said that so we said this and I'm just explaining this and in this session my objective is to give you a proof of it. So first of all let us understand what we understand in the previous session. So previous session we talked about if px is ax square plus bx plus c which is a quadratic polynomial you can see that. Now we are interested in finding the sign of it, sign means whether px is px will be greater than 0 or px will be less than 0 can be predicted so the idea is can I predict this thing whether it is greater than 0 or less than 0 on the basis of only one thing that is the coefficient of x square right and we said that px will be having the sign of a px will have same sign of a same sign same sign as that of a when I say px it means for any value of a x if you take any x value for example x is equal to say x equals to minus 5 x equals to root 2 and you try to evaluate px for all these okay so when you do that you will be you know you without even evaluating the px values for all these you know x values you can predict it whether it px is going to be greater than 0 or less than 0 just by seeing the value of a that's what we said and we said that px will be greater than 0 if x is greater than sorry a is greater than 0 and px is going to be less than 0 if a is going to be less than 0 right but it fails it fails at one this particular concept fails when so the exception I'm writing the exception is if by chance px equals to 0 has real and equal roots unequal sorry real and unequal roots real and unequal roots then this doesn't hold and in fact exactly opposite holds so hence px will be greater than 0 px is greater than 0 if a is less than 0 and px is less than 0 if a is greater than 0 now has real and unequal roots but one thing I just missed out was that that that those real roots let's say those real roots are alpha and beta and you are picking up any x such that x is less than alpha sorry less than beta and it is greater than alpha so basically x is lying between alpha and beta if you pick some x like that right and then alpha and beta real then the sign of px and a will be exactly opposite that's what we meant now let us prove it this is what we did in the last session if you want to understand this more you can just check out the last session so let us take the proof of it how do I prove it so prove this very simple so let us take case by case so case one what is this case this case is let us say alpha and beta are are real okay and what are alpha beta real and unequal unequal roots of ax plus bx plus c equals to 0 okay this is a case one right so we are taking when alpha and beta are real and unequal in case 2 will take when alpha and beta real and equal and case 3 will take alpha beta unreal non real that is complex so case by case we are going to take this up so let us say alpha and beta are real and unequal roots of ax square plus bx plus c and let us assume that alpha is less than beta right so if they are real and unequal one of them definitely will be less than the other so let us say alpha is less than beta okay so let me write this expression so ax square plus bx plus c can be written as if I take a common I can write this as x square plus b upon ax plus c upon a okay this is what I can reduce it to right now we learned in the previous sessions that if you have if you have an equation ax square plus bx plus c equals 0 and alpha and beta are roots of this a root then sum of the root alpha beta is minus b by a and product of roots is nothing but c by a okay and hence the polynomial can be expressed as x square times or basically you can write this as a times x square plus or not not plus actually minus b by a minus minus b by ax plus c by a is equal to 0 can be written as so if you see this is what I'm trying to ascertain yep so this is the same this equation is same as this so hence can I not write the given polynomial here now I'm coming to the polynomial here so this can be written as a times x square minus minus b by ax plus c by a and hence it can be written as a x square minus minus b by a is alpha plus beta so hence you write alpha plus beta see minus b by a is alpha plus beta x and here you can write alpha beta correct so hence you will get a and then what open the brackets you'll get x square minus alpha x minus beta x plus alpha beta right which is equal to a x common x minus alpha minus beta times x minus alpha again so hence I could factorize this as a times x minus alpha and x minus beta right now if if alpha if let's say now you are you're trying to find out px value so px is given as a times x minus alpha times x minus beta right now if if x let's say is less than alpha if x is less than alpha and alpha any was beat less than beta so x will be less than let us say we are considering this type this case so what will happen x minus alpha is negative or let me write this as x minus alpha will be less than 0 similarly x minus beta will be less than 0 so hence x minus alpha x minus beta both together will be greater than 0 why because this is negative negative negative times negative is positive right so px is equal to a quantity a into a positive quantity positive quantity that means if a is positive then px is positive if a is positive and what positive multiplied by positive is positive and if a is less than or if a is negative then px is also negative right so it matches right so hence px and a have same sign same sign just by looking at a you can predict the sign of px now you will say okay this is for the one case where x is less than alpha less than beta what if so other case will be alpha is less than beta and let's say beta is also less than x okay so now what x minus alpha will be greater than 0 positive because alpha is less than x x minus beta will be also greater than 0 because beta is also less than x so if you subtract a smaller quantity from a bigger quantity you'll get a positive quantity that means x minus alpha times x minus beta is greater than 0 two positive quantities multiplied together will give you this that means now if you write px again you can write as px is a times positive quantity positive quantity that means again you can infer that px and a have same sign isn't it so if px is positive a is positive then px will be positive and if a is negative then px will be negative now fair enough but what if x is in between alpha and beta then what will happen x minus alpha is greater than 0 clearly but x minus beta will be less than 0 x is smaller than beta so hence x minus alpha and x minus beta will be less than 0 so hence px as we found out was equal to a times x minus alpha times x minus beta so hence I can say this is nothing but a times a negative quantity negative quantity so hence px will be greater than 0 only when a is less than 0 is negative only then negative times negative will become positive isn't it and px is going to be less than 0 if a is greater than 0 right that means if x lies between alpha and beta then px will have opposite sign opposite sign that of a clear so this case one so case one we took three cases again where alpha and beta x is less than alpha and beta they're also alpha and x is greater than alpha and beta and the third case is x is between alpha and beta in all the three cases we could determine what will be the sign of px depending upon sign of a now case two would be when roots are real and equal roots of what of equation ax square plus dx plus c equals to 0 if the roots are real and equal let the root be root be let the root be alpha only so hence what we can say alpha and alpha so two roots and equal right so px will be clearly what a times x minus alpha times x minus alpha which is nothing but a times x minus alpha squared right now x minus alpha squared is always greater than 0 if alpha is not 0 if x is not alpha then x minus so if x is not equal to alpha then x minus alpha squared will always be greater than 0 why because these are square quantities squares are always positive square quantity right always positive there's no square which is negative so hence we can say px is going to be a times a positive quantity isn't it now the moment we say that again we get the result what px and a have same sign isn't it so if a is greater than 0 px is going to be greater than 0 right because a times positive quantity will be a positive quantity and if a is less than 0 then px again is going to be less than 0 because negative times are positive quantity is always negative so hence that's established this is case 2 now case 3 would be case 3 roots are imaginary or roots are let's say complex or we say non real roots are roots are not real roots are not real okay then what will happen if roots are not real we know that what happens if roots roots are not real then what we can say we can say this that px is equal to again I am saying again I am saying a times now I am writing this px as yeah so px is a times x square plus b by ax plus c by a isn't it so hence I can say this as a and here I can write x square plus 2 times b by or 2 times 2 times x upon b by 2 a plus b square by 4 a square it looks like I am trying to complete the square then I have then I have c by a and minus b square by 4 a square okay I can do that so this 2 times b by 2 a plus b square by 4 a square 2 terms cancel each other out and this 2 and this 2 cancels each other out to get this back isn't it so hence what I am trying to say is this particular expression is same as this expression why did I do this let's see a times I can now say x plus b by 2 a whole square plus b by 2 a whole square isn't it and I can say here it is nothing but 4 a c minus b square by 4 a square 4 a square isn't it I just took the common denominator can I say that yes I can now guys roots are not real what is that condition when a roots not real so roots are not real when d is less than 0 you know this and what was d b square minus 4 a c is less than 0 and b square minus 4 a c is less than 0 of any quadratic equation which equation this equation a x square plus b x plus c equals to 0 if b square minus 4 a c is less than 0 then roots are not not real right so hence b square minus 4 a c is less than 0 so hence 4 a c minus b square will be greater than 0 clearly isn't it 4 a c minus b square so hence if this is if take negative sign negative of it so negative of a negative number is always positive right so hence now observe p x was given as a times a positive number first term is positive see this term is going to be positive why because it is a square and second term is also positive why because 4 a c minus b square is a positive quantity and divide by a positive quantity again square is always a positive quantity so hence this entire term is greater than 0 isn't it so that means now if this in this which is greater than 0 this term not entire term because a is also being multiplied so I can't write this expression here but I can definitely write this that a positive quantity plus a positive quantity isn't it both which both I'm talking about this quantity is positive clearly because it's a square and this also is a positive quantity why because the numerator is positive which is given here and denominator is any major square so positive so hence again we can ensure that p x and a have same sign positive and positive negative and negative now if you analyze all the inferences so this was one inference this was one inference when complex and real roots were there when equal and real roots also the inference was this p x and a have same sign and when case 1 was there that is real and unequal real unequal in that case also we had same sign p x and a have same sign p x and a have same sign but there was only one exception this that if they are real and unequal and if by chance x is lying between alpha and beta the two roots then p x and a will be of opposite sign in all other cases for any value of x and b x and b x and b x and a will be of opposite sign in all other cases for any value of x p x and a will be of the same sign that means if you know the sign of a and if you know where x is you can predict the sign of p x without calculating the value of p x this is what this exercise was all about so we could prove this okay so hence summary once again I can write what is the summary p x is equal to a x square plus p x plus c will have same sign as that of a except except p x is equal to 0 has real and unequal roots and x lies between the two roots