1.10: Transform the roots of a quadratic

4PM1/1R/November/2020 — Question 8 · 7 marks

f(x)=3x2−x+4,g(x)=x2−px+q.f(x)=3x^2-x+4, \qquad g(x)=x^2-px+q.
The roots of the quadratic equation f(x)=0\displaystyle f(x)=0 are α\displaystyle \alpha and β\displaystyle \beta.
The roots of the quadratic equation g(x)=0\displaystyle g(x)=0 are
(α+1α)and(β+1β).\left(\alpha+\frac1\alpha\right) \qquad \text{and} \qquad \left(\beta+\frac1\beta\right).
Without solving the equation f(x)=0\displaystyle f(x)=0,
(a) show that
p=712,p=\frac{7}{12},
(3)
(b) find the value of q\displaystyle q.
(4)