1.34: Quadratic Roots and a Transformed Equation

4PM1/2/January/2019 — Question 8 · 11 marks

The roots of the equation
3x2−2x−1=03x^2-2x-1=0
are α\displaystyle \alpha and β\displaystyle \beta, where α>β\displaystyle \alpha>\beta.
Without solving the equation,
(a) find the value of α2+β2\displaystyle \alpha^2+\beta^2.
(3)
(b) Show that
α−β=43.\alpha-\beta=\frac43.
(2)
(c) Form a quadratic equation, with integer coefficients, that has roots
α+βαandα−ββ.\frac{\alpha+\beta}{\alpha} \qquad \text{and} \qquad \frac{\alpha-\beta}{\beta}.
(6)