1.22: Past-paper question 10

4PM1/1R/June/2024 — Question 10 · 12 marks

The quadratic equation 2x2+kx+4=0\displaystyle 2x^2 + kx + 4 = 0 has roots α\displaystyle \alpha and β\displaystyle \beta such that
k<0andα>βk < 0 \mathrm{and} \alpha > \beta
Given that α2−β2=7174\displaystyle \alpha^2 - \beta^2 = \frac{7\sqrt{17}}{4}
(a) show that k=−7\displaystyle k = -7
(8)
(b) Hence form a quadratic equation that has roots
(α−β)and(α+β)(\alpha - \beta) \mathrm{and} (\alpha + \beta)
(4)