If `alpha,beta,gamma,delta` are the roots of the equation `x^4-K x^3K x^2+L x+m=0,w h e r eK ,L ,a n dM` are real numbers, then the minimum value of `alpha^2+beta^2+gamma^2+delta^2` is `0` b. `-1` c. `1` d. `2`

Options

A. 0
B. -1
C. 1
D. 2

Answer

2

Explanation

Roots of equation `x^(4) - kx^(3) + Kx^(2) + Lx + M = -0` are `alpha, beta, gamma, delta`
`sumalpha = K, sumalphabeta = K,`
`sumalphabetagamma = L, alphabetagammadelta = M`
`alpha^(2) + beta^(2) + gamma^(2) + delta^(2) = (alpha + beta + gamma + delta)^(2) - 2sumalphabeta`
`K^(2) - 2K`
`= (K-1)^(2) - 1`
`therefore (alpha^(2) + beta^(2) + gamma^(2) + delta^(2))_("min") = -1`