IAL 2019 May FP1 Q6

Question

[!problem]

The quadratic equation

$2x^2 + x + 4 = 0$

has roots $\alpha$ and $\beta$.

Without solving the quadratic equation,

(a) write down the value of $(\alpha + \beta)$ and the value of $\alpha\beta$.

(b) find the value of

(i) $\alpha^2 + \beta^2$

(ii) $\alpha^4 + \beta^4$

(c) find a quadratic equation that has roots

$\frac{1}{\alpha^2} \text{ and } \frac{1}{\beta^2}$

giving your answer in the form $px^2 + qx + r = 0$, where $p$, $q$ and $r$ are integers.

(4)