1.31: Algebraic Relations between Quadratic Roots

4PM1/2/June/2019 — Question 10 · 15 marks

The roots of the equation
x2+3x−5=0x^2+3x-5=0
are α\displaystyle \alpha and β\displaystyle \beta.
(a) Without solving the equation, find
(i) the value of α2+β2\displaystyle \alpha^2+\beta^2,
(ii) the value of α4+β4\displaystyle \alpha^4+\beta^4.
(5)
Given that α>β\displaystyle \alpha>\beta and without solving the equation,
(b) show that
α−β=29.\alpha-\beta=\sqrt{29}.
(2)
(c) Factorise α4−β4\displaystyle \alpha^4-\beta^4 completely.
(3)
(d) Hence find the exact value of α4−β4\displaystyle \alpha^4-\beta^4.
(2)
Given that
β4=p+q29,\beta^4=p+q\sqrt{29},
where p\displaystyle p and q\displaystyle q are positive constants,
(e) find the value of p\displaystyle p and the value of q\displaystyle q.
(3)