If p and q are the roots of the quadratic equation $\displaystyle x^{2}-\left ( \alpha -2 \right )x-\alpha -1=0$. What is the minimum possible value of $\displaystyle p^{2}+q^{2}$?

By Vieta's formulas for the equation x^2 - (alpha - 2)x - (alpha + 1) = 0, the sum of the roots p and q is p + q = alpha - 2. We want to find the minimum of p^2 + q^2, which can be rewritten using the identity p^2 + q^2 = (p + q)^2 - 2pq. The product of the roots is pq = -(alpha + 1), so substituting the sum and product gives (alpha - 2)^2 - 2(-(alpha + 1)). Expanding this expression yields alpha^2 - 4alpha + 4 + 2alpha + 2, which simplifies to the quadratic alpha^2 - 2alpha + 6. To find the minimum of this quadratic in terms of alpha, we complete the square to get (alpha - 1)^2 + 5. Since the square of a real number is always non-negative, the minimum possible value is 5, occurring when alpha = 1.

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