Roots of quadratic equations

Sample questions

Original practice questions written in the style of the exam.

1. Question 15 marks

   The roots of $x^2 - 4x + 7 = 0$ are $\alpha$ and $\beta$.

   (a) Without solving the equation, find the value of $\alpha^3 + \beta^3$.

   (b) Find a quadratic equation with integer coefficients whose roots are $\dfrac{\alpha}{\beta}$ and $\dfrac{\beta}{\alpha}$.

   Mark scheme

   • B1 $\alpha + \beta = 4$, $\alpha\beta = 7$ (1)
   • A1 ${\alpha }^3+{\beta }^3=(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta )$ (1)