If $$\alpha$$ and $$\beta$$ are the roots of x² + x + 1 = 0, then what is $$\sum\limits_{j = 0}^3 {({\alpha ^j} + {\beta ^j})} $$ equal to?

Under which one of the following conditions will be quadratic equation x² + mx + 2 = 0 always have real roots?

If both p and q belong to the set {1, 2, 3, 4}, then how many equations of the form px² + qx + 1 = 0 will have real roots?

What is the value of k for which the sum of the squares of the roots of $$2{x^2} - 2(k - 2)x - (k + 1) = 0$$ is minimum?