let m,n be the roots of (x^2)-x+p=0 and u,v be the roots of (x^2)-4x+q=0. If m,n,u,v are in G.P. then the integral values of p &q are ?

as m,n are roots of first equation, m*n = p. similarly u*v = q. also m+n = 1 and u+v = 4. [I think you are aware that for an equation a*x^2 + b*x +c=0 sum of roots = -b/a and product of roots is c/a] as m,n,u,v are in GP, let n = m*r, u = m*r^2, v = m*r^3. ==> m*(1+r) = 1 and m*(1+r)*r^2 = 4 [just substitution] ==> r^2 = 4. ==> r = 2 or -2. if r=2, m = 1/3 . then p = m*n = (1/3)*(1/3)*2 = 2/9 which is not an integer. so r = -2. ==> m=-1. therefore p = (-1)*(-1)*(-2) = -2, q = u*v = m^2*r^5 = -32