1.11: Equations whose roots are transformations of quadratic roots

4PM1/2/November/2020 — Question 6 · 11 marks

f(x)=4x2−3x−5.f(x)=4x^2-3x-5.
The equation f(x)=0\displaystyle f(x)=0 has roots α\displaystyle \alpha and β\displaystyle \beta.
Without solving the equation f(x)=0\displaystyle f(x)=0,
(a) form an equation, with integer coefficients, that has roots
2αβand2βα.\frac{2\alpha}{\beta} \qquad \text{and} \qquad \frac{2\beta}{\alpha}.
(6)
g(x)=4x2+px+q,g(x)=4x^2+px+q,
where p\displaystyle p and q\displaystyle q are constants.
Given that the equation g(x)=0\displaystyle g(x)=0 has roots 3α+β\displaystyle 3\alpha+\beta and α+3β\displaystyle \alpha+3\beta,
(b) find the value of p\displaystyle p and the value of q\displaystyle q.
(5)