Multiple choice

If $\alpha, \beta, \gamma,\delta$ are the roots of the equation $x^4 - Kx^3 + Kx^2 + Lx + M = 0$, where $K, L$ & $M$ are real numbers, then the minimum value of $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$ is

  1. $0$
  2. $-1$
  3. $1$
  4. $2$
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B Correct answer
Explanation

The sum of the squares of the roots is given by (sum of roots)^2 - 2(sum of roots taken two at a time). For x^4 - Kx^3 + Kx^2 + Lx + M = 0, the sum of roots is K and the sum of roots taken two at a time is K. Thus, the sum of squares is K^2 - 2K, which is (K-1)^2 - 1. The minimum value is -1 when K = 1.

AI explanation

Using Vieta's formulas for x^4 - Kx^3 + Kx^2 + Lx + M = 0, the sum of the roots alpha + beta + gamma + delta equals K. We can express the sum of the squares of the roots as (alpha + beta + gamma + delta)^2 - 2(alpha*beta + alpha*gamma + ...), which simplifies to K^2 - 2K. To find the minimum value of this quadratic expression K^2 - 2K, we complete the square to get (K - 1)^2 - 1, showing the minimum possible value is -1.

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