1.32: Quadratic Roots with Parameters

4PM1/2R/June/2019 — Question 11 · 12 marks

The quadratic equation
x2−px+q=0,x^2-px+q=0,
where p>0\displaystyle p>0, has roots α\displaystyle \alpha and β\displaystyle \beta.
Given that
2αβ=32\alpha\beta=3
and that
4(α2+β2)=k2−6k−3,4(\alpha^2+\beta^2)=k^2-6k-3,
where k>3\displaystyle k>3,
(a)
(i) write down the value of q\displaystyle q,
(ii) find an expression, in terms of k\displaystyle k, for p\displaystyle p.
(5)
Given also that
7αβ=3(α+β),7\alpha\beta=3(\alpha+\beta),
(b) find the value of k\displaystyle k.
(2)
(c) Hence form an equation, with integer coefficients, which has roots
αα+βandβα+β.\frac{\alpha}{\alpha+\beta} \qquad \text{and} \qquad \frac{\beta}{\alpha+\beta}.
(5)