Given that the roots of 2x^2-4x+5 are alpha and beta,find the value of ...

Question

(i)alpha^3beta^3
(ii)(alpha-beta)^2

Answer 1

Using the sum and product of roots property, we know
a+b = - (-4/2) = 2 and ab = 5/2, where a and b are alpha and beta

we know that
(a+b)^3 = a^3 + b^3 + 3a^2b + 3ab^2
(a+b)^3 = a^3 + b^3 + 3ab(b+a)
2^3 = a^3 + b^3 + 3(5/2)(2)
8 = a^3 + b^3 + 15
a^3 + b^3 = -7

in the same way....
(a-b)^2 = a^2 - 2ab + b^2
= a^2 +b^2 - 2ab

but a^2 + b^2 = (a+b)^2 - 2ab

so
(a=b)^2 = (a+b)^2 - 2ab - 2ab
= 4 - 5/2 - 5/2
= -1

Answer 2

in the third last line I should have had

(a-b)^2 = (a+b)^2 - 2ab - 2ab