Question

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - ax - b = 0 \) with \( \text{Im}(\alpha) <\text{Im(\beta) \). Let \( P_n = \alpha^n - \beta^n \). If} \[ P_3 = -5\sqrt{7}, \, P_4 = -3\sqrt{7}, \, P_5 = 11\sqrt{7}, \, P_6 = 45\sqrt{7}, \] then \( |\alpha^4 + \beta^4| \) is equal to:

Hint:

In problems involving powers of roots of a quadratic equation, use recurrence relations to compute higher powers and solve for desired expressions.

Answer 1

Correct Answer: 31

We are given the following relations: \[ P_n = \alpha^n - \beta^n, \] and the values: \[ P_3 = -5\sqrt{7}, \, P_4 = -3\sqrt{7}, \, P_5 = 11\sqrt{7}, \, P_6 = 45\sqrt{7}. \] We need to find \( |\alpha^4 + \beta^4| \). We know from the given quadratic equation \( x^2 - ax - b = 0 \) that: \[ \alpha + \beta = a \quad \text{and} \quad \alpha \beta = -b. \] Using the recurrence relation for \( P_n \) and the known values of \( P_3, P_4, P_5, P_6 \), we compute \( \alpha^4 + \beta^4 \).