Question

I need help on part b the showing part!!

Answer 1

X^2 = 3x + 5
x^2 - 3x - 5 = 0

a(alpha) and b(beta are roots
(x - a)(x - b) = 0
x^2 - (a+b)x + ab = 0

compare coefficients
a+b = 3
ab = -5

solving
(1/a^2 + 1/b^2)
= (a^2 + b^2)/(ab)^2
= (a^2 + 2ab + b^2 - 2ab)/(ab)^2
= [(a+b)^2 - 2ab]/(ab)^2
= [(3)^2 - 2(-5)]/(-5)^2 =...

prove
a^4 = 57a + 70

let x=a,
a^2 = 3a + 5
(a^2)^2 = (3a + 5)^2
a^4 = 9a^2 + 30a + 25
a^4 = 9(3a + 5) + 30a + 25
a^4 = 57a + 70