1.20: Past-paper question 10

4PM1/2/November/2023 — Question 10 · 10 marks

The roots of a quadratic equation are α\displaystyle \alpha and β\displaystyle \beta where
α+β=−52andα3+β3=1158\alpha + \beta = -\frac{5}{2} \quad \mathrm{and} \quad \alpha^3 + \beta^3 = \frac{115}{8}
(a) Show that αβ=4\displaystyle \alpha\beta = 4
(3)
(b) Form a quadratic equation with integer coefficients, that has roots
α2+1βandβ2+1α\frac{\alpha^2 + 1}{\beta} \quad \mathrm{and} \quad \frac{\beta^2 + 1}{\alpha}
(7)