) = x + x – . By the method of splitting the middle term, x + x – = x + x – x – = x ( x + ) – ( x + ) = ( x – )( x + ) Hence, the value of x + x – is zero when either x – = or x + = , i.e., when x = or x = – . So, the zeroes of x + x – are and – . Observe that : Sum of its zeroes ) Product of its zeroes = In general, if * and * are the zeroes of the quadratic polynomial p ( x ) = ax + bx + c , a , then you know that x – and x – are the factors of p ( x ).
Therefore, ax + bx + c = k ( x – ) ( x – ), where k is a constant = k [ x – ( + ) x + ] = kx – k ( + ) x + k Comparing the coefficients of x , x and constant terms on both the sides, we get a = k , b = – k ( + ) and c = k This gives + = –b a , = c * , are Greek letters pronounced as ‘alpha’ and ‘beta’ respectively. We will use later one more letter ‘ ’ pronounced as ‘gamma’. i.e., sum of zeroes = + = ) product of zeroes = = c Let us consider some examples. Example : Find the zeroes of the quadratic polynomial x + x + , and verify the relationship between the zeroes and the coefficients.