if the roots of equation ax²+bx+c=0 are of form alpha/alpha-1 &alpha+1/alpha then value of (a+b+c)²is

let roots be p/(p-1) & (p+1)/p

product of roots (p+1)/(p-1) = c/a

so p=(c+a)/(c-a)  &  ..................(i)              (by componendo-dividendo)

2/(p-1)=(c-a)/a    .................(ii)                 " "    "     "      "     "      "   " "

multiplying i & ii

p/(p-1) = (c+a)/2a   now as it is root , it must satisfy the given eqn

so   a(c+a)²/4a²   +   b(c+a)/2a  +  c =0

or b² + (c+a)² +2b(c+a) + 4ac = b²

or   (a+b+c)² = b²-4ac