 Now in this question, we have to find out the zeros of the polynomial fx equals abx square plus b square minus acx minus bc So this is a given quadratic polynomial. So coefficient of x square you can see this is ab Coefficient of x is b square minus ac and the constant term is minus bc, isn't it? And we have to find the roots and verify the relationship between the zeros and the coefficients, right? So how to find out the zero so you know Finding out zero is nothing but finding those values of x which make fx zero. So we are saying fx is equal to zero We have to find those values of x for which x is zero Okay, so let us write abx squared minus So it is plus b square minus acx minus bc equals zero now if you know the quadratic formula or Sridhara Sridhara Charya's Rule you can always find that but we will try and see if we can split the middle term now if you see the product of the coefficient of x square and the constant term is nothing but ab times minus bc Which is minus ab square c, isn't it? Now the same thing can also be seen as b square minus ac if you see these are the two You know the split terms if I multiply this also you'll get the same thing so hence and thankfully it's given in the question itself, right? so now We can write this particular equation as abx square plus b square x Minus acx minus bc equals zero Now taking bx common from the first two term you'll get ax plus x sorry ax plus b is it yeah and Taking c common minus c common you'll get ax plus b again. This is equal to zero So hence this can be written as ax plus b times bx minus c is equal to zero that means ax plus b is equal to zero or bx minus c equals to zero that means x is equal to minus b upon a or x is equal to Plus c upon b so the zeros of the quadratic polynomial here is the zeros are zeros zeros are alpha let us say alpha is equal to minus b by a and Beta is equal to c by b So alpha plus beta will be is equal to nothing but minus b by a plus c by b Which is nothing but minus b square plus ac by a b Is it it and now let us also write the if you see the given quadratic polynomial was this so here the b term the coefficient of x is this much and coefficient of a b is this So hence sum of roots from there will be nothing but minus coefficient of x upon coefficient of x squared isn't it so if you see alpha plus beta is minus coefficient of x divided by coefficient of x squared So what is coefficient of x guys if you see coefficient of x is this much here So b square minus ac and this minus sign is because there was a minus sign here and divided by coefficient of a x square Which is a b right so I divided that and if you see we get the same as The left hand side so this is a b now if you see Both by actually summing the roots and from the formula of some of roots we get the same thing so hence verified So some of roots very fine isn't it? Now let us say what is alpha times beta that is product of roots product of roots is nothing but minus b by a the first Roots I'm not I'm not supposed to use the word roots product of zeros So minus b by a into c by b is nothing but minus c by a This is the product of the zeros now product of the zeros from the expression is alpha beta is given as nothing but constant term constant term divided by coefficient of x square isn't it so what is constant term guys let's check constant term was minus BC right so constant term was minus BC and the coefficient of x square was a b so it's nothing but minus c upon a so again if you see Both are same Right both are same so one we found out the actual product of the zeros and the other we found out the zeros the product of zeros from The the polynomial itself without actually doing anything or without solving the equation or without finding the actual zeros both Match so hence here also it is verified Product of roots also verified