First slide

Theory of equations

Question

If $α, β, γ, δ$ are the roots of the equation $x^{4} - {Kx}^{3} + {Kx}^{2} + Lx + M = 0$ where K, L, and M are real numbers, then the minimum value of $α^{2} + β^{2} + γ^{2} + δ^{2}$ is

Moderate

A

0
B

-1
C

1
D

2

Solution

Roots of equation $x^{4} - {Kx}^{3} + {Kx}^{2} + Lx + M = 0 are α, β, γ, δ$
$\begin{array}{l} Σα = K, Σαβ = K \\ Σαβγ = - L, αβγδ = M \\ α^{2} + β^{2} + γ^{2} + δ^{2} = (α + β + γ + δ)^{2} - 2 Σαβ \\ = K^{2} - 2 K \\ = (K - 1)^{2} - 1 \\ ∴ {(α^{2} + β^{2} + γ^{2} + δ^{2})}_{min} = - 1 \end{array}$

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