 Hello and welcome to the session. Let us discuss the following problem today. Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients. We have the polynomial f of s is equal to 4s square minus 4s plus 1. Now let us understand the key idea for the question. A real number alpha is a zero of a polynomial f of x if f of alpha is equal to 0 and if alpha and beta are the coefficients of a quadratic polynomial f of x is equal to a square plus bx plus c then alpha plus beta is equal to minus b by a which is equal to minus coefficient of x by coefficient of x square and alpha beta is equal to c by a which is equal to constant term by coefficient of x square. Now let us write the solution for the question. We have f of s is equal to 4s square minus 4s plus 1. Now we split the middle term we get 4s square minus 2s minus 2s plus 1. Now we take the common factors from this two we take 2s common so we get 2s minus 1 and from this two we take minus 1 common so we get 2s minus 1. So we get 2s minus 1 into 2s minus 1 which is our f of s. Now zeros of f of s are given if f of s is equal to 0. So therefore 2s minus 1 into 2s minus 1 is equal to 0 which implies 2s minus 1 is equal to 0 or 2s minus 1 is equal to 0 which implies s is equal to half or s is equal to half thus zeros of f of s are alpha is equal to half and beta is equal to half. Now the second part of our question that is to verify the relationship between coefficients and the zeros. Sum of zeros is equal to alpha plus beta which is equal to half plus half which is equal to 1 that is minus coefficient of x divided by coefficient of x square which is equal to minus of minus 4 by 4 which is equal to 1 by 1. Similarly product of zeros which is equal to alpha beta which is equal to half into half which is equal to 1 by 4 that is constant term divided by coefficient of x square which is equal to 1 by 4 hence verified. Hope you understood the problem. Bye and have a nice day.