 Hi friends welcome to this session on quadratic equation now here is a deadly looking statement but don't get frightened by it you will see that it is very easy to understand once the explanation is done but nevertheless in the first impression it is very very dangerous looking statement but what does it say let's try to break it down into bits and pieces so it says for all real values of x so mind you x is real the expression a x square plus b x plus c clearly it is a polynomial quadratic polynomial has the same sign as a okay so let's concentrate in this part so they are saying for all real values of x the expression a x square plus b x plus c has the same sign as a so whatever is the sign of a px will have the same sign that means if a is positive then px also will be positive if you calculate evaluate the value of px it will also be positive and if a is negative if a is negative then px also is negative px is px also is negative except but there is some exception and the exception is when the roots of the equation roots of the equation this let us say the roots are alpha and beta are real and unequal that means roots of the equation are real and unequal and x has a value lying between them that means the this part of the statement says if alpha is less than x less than beta if this happens where alpha and beta are the roots of the equation which equation this equation that means you are equating px to zero and finding the roots alpha and beta or the zeros of the polynomial px alpha and beta are zeros of the polynomial px or roots roots of px equals to zero if you find them and x lies between alpha and beta then px will have sign opposite to a right once again what does it mean that if a is positive and a is if a is positive then px will also be positive if a is negative then px will also be negative except for the case when x lies between alpha and beta where in alpha and beta the roots of px equals to zero quadratic equation then this will not be followed exactly opposite will happen what is opposite if px if a is positive if a is positive then px will be negative and vice versa if a is negative then px will be positive okay let's take an example to ease out the tension which we had got you know in this discussion so what I'm saying is let us say we have this let's take an p let's take a px px is let's say x square minus 3x plus 2 okay clearly if you solve it I'm not going to solve this equation but if you solve it you'll get alpha is equal to 1 and beta is equal to let's say 2 okay so 1 and 2 are the roots of this equation so except you know and here if you see there are real roots yeah if they are not no real roots then no problem at all then but if they're real if you know so let's evaluate so what is a here a is 1 so px will be positive right because a is greater than 1 always but but for the case where if x is less than alpha less than sorry greater than alpha and less than beta right once again once again I'm trying to explain it a little slowly please understand what I'm saying is px will be positive if a is greater than 0 similarly px will be negative if a is less than 0 except where where if roots of px equals to 0 are real and unequal and x lies between the two roots in this that case px will be greater than 0 if a is less than 0 and px will be less than 0 if a is greater than 0 okay so let's take this example again so example is px is x square minus 3x plus 2 and this is px right so for all values of x for all values of x for all x px will be greater than 0 why because a here is 1 which is greater than 0 right a is 1 can you see this right but but but for few exceptions if x lies between 1 and 2 right where 1 alpha is equal to 1 and beta is equal to 2 are real and unequal roots real and unequal roots of unequal roots of px equals to 0 and x is lying between 1 and 2 okay then what will happen in the in such circumstances px is less than 0 right why because a anyway is greater than 0 so hence px will be less than 0 let's try some values so let's say x is equal to 0 x equals to 0 it definitely doesn't lie in the interval 1 less than x less than 2 right so hence hence what will happen px must be greater than 0 let's check putting x equals to 0 in the given expression so 0 square minus 3 times 0 plus 2 this is what we get so basically p0 will do and clearly it is 2 which is greater than 0 which is positive okay let us take x equals to 3 again x equals to 3 doesn't lie in this given interval correct so hence we can predict the sign of p3 p3 will be greater than 0 without even calculating I can say why because you know I know a is greater than 1 sorry a is greater than 0 so if a is positive px has to be positive if it doesn't lie between alpha and beta x doesn't lie between alpha and beta so x clearly doesn't lie between 1 and 2 so hence it will be positive let's check so 3 square minus 3 times 3 plus 2 which is nothing but 11 minus 9 which is 2 again so if you see this is also greater than 0 right so here it's working but let us check by putting x equals to 1.5 which is between so clearly one less than 1.5 less than 2 it is lying between 1 and 2 the roots of px equals to 0 then here p 1.5 will be I am saying p 1.5 is greater going to be less than 0 why because 1.5 lies between 1 and 2 and my a is 1 which is positive so hence p of x will be negative let's try so p 1.5 is nothing but 1.5 square minus 3 times 1.5 plus 2 which is nothing but 2.25 minus 4.5 plus 2 which is equal to 0.25 but negative see so p of 1.5 is coming out to be negative right and it matches our prediction as well so hence it is exactly opposite of sine of a so whatever is the sine of a px has a opposite sign right so this is what it means let us take another example let us say I have px is equal to minus 2x square minus 3x and plus let's say you know 2 plus 2 yeah so this is px right so can clearly you can see a is less than what a is less than 0 so a is negative a is negative so hence px will be negative px is going to be negative but exceptions exceptions px is going to be negative for all x values for all x values but exception what is the exception if px equals to 0 has real real roots real and equal sorry unequal roots unequal roots then px will have px will be greater than 0 px will be greater than 0 for all x lying between alpha less than x less than so between alpha and beta if x is there then px is greater than 0 in this case why because a is negative so whatever so px matches the sine of a until unless the value of x lie between the two roots this is the summary so let's see okay so px what are the roots let's try to find them out first so if I take minus 2 common so minus common so 2x square plus 3x minus 2 right so hence it is minus 2x square plus 4x minus x minus 2 correct so this is minus 2x square plus so you can take 2x common so hence if you take 2x common it will be x plus 2 minus x plus 2 right so hence it is nothing but minus x plus 2 times 2x minus 1 okay this is my yeah so this is the factorization so what are the roots so alpha clearly is equal to 1 and beta is minus 2 or whichever way alpha is minus 2 hence so since I am taking alpha as to be a smaller one so let's let's call beta as 1 beta as 1 so beta is equal to 1 and alpha is minus 2 so hence the roots are real and unequal so hence px will be negative for all x negative why negative because it has to match with the sign of a right so but sign of a here is minus minus so hence px will be matching that sign for all x but px will be greater than 0 for all x which which type of x minus 2 less than x less than 1 let's check it so if you take x equals to let's say 2 what will be px px is clearly minus 2 plus 2 and 2 into 2 minus 1 I'm taking this expression for calculation of px which is equal to minus 4 times 3 which is minus 12 which is clearly less than 0 which is sign of a isn't it it matches with sign of a here a is minus 2 right let us take x equals to minus 3 okay so px will be minus of minus 3 plus 2 and 2 times minus 3 minus 1 okay so what is this value it is nothing but minus minus 3 plus 2 is minus 1 isn't it and here it is 2 into minus 3 is minus 6 minus 1 so which is clearly minus minus minus 7 isn't it so which is clearly less than 0 or it matches with sign of a sign of a which is negative so px is matching with the sign of a until what condition let us take x equals to 0 and clearly 0 is less than or greater than minus 2 and less than 1 okay so if you put p0 what will you get p0 is if you put 0 here so minus x plus 2 and so minus 0 plus 2 times 2 times 0 minus 1 which is nothing but 2 into so p0 is 2 minus x plus 2 so 0 plus 2 and 2 times 0 minus 1 so hence it is nothing but my so there's a minus sign here so it will be minus 2 into minus 1 which is equal to 2 is greater than 0 so hence if you see it is positive which is opposite to sign of a since a is negative which is negative isn't it a is negative so hence what do we infer px matches sign of px sign of px matches with sign of a right except except when if px is equal to 0 has real and unequal root alpha beta and alpha less than x less than beta okay so px will always match the sign of a except when if px equals to 0 has real and unequal roots alpha beta and you are trying to evaluate px for those values of x which is less than alpha greater than alpha and less than beta right so I hope you could understand this concept in the next session we'll try to prove this