Question

the equation (x-1)(x-3)=k has both roots lying between 1 and 3, then number of integral values of k is

Answer 1

since the vertex of (x-1)(x-3) = 0 is at (2,-1), k ≥ -1

or, since
x^2 - 4x + 3-k has discriminant 16-4(3-k) = 4(1+k),
1+k must not be negative