1.14: Relationships between roots

4PM1/2/June/2023 — Question 11 · 12 marks

The roots of a quadratic equation

E

are

\alpha

and

\beta

where

\alpha > \beta > 0

Given that

\alpha - \beta = 2\sqrt{6}

and

\alpha^2 + \beta^2 = 30

(a) show that

(i) \alpha\beta = 3

(4)

(ii) \alpha + \beta = 6

(2)

(b) Without solving

E

(i) \text{ find the value of } \alpha^4 + \beta^4

(2)

(ii) \text{ find the exact value of } \alpha^4 - \beta^4

(2)

Given that

\alpha^4 = P + Q\sqrt{6}

where

P

and

Q

are positive integers,

(c) find the value of

P

and the value of

Q

(2)