Question

if alpha and beta are the zeros of x^2+x+5 then find
(a) 1/alpha +1/beta
(b) alpha^2 + beta^2
(c) alpha^2 - beta^2

Answer 1

(a) from the equation alpha+beta= -1 and (alpha)*(beta)=5 .1/alpha+1/beta=alpha+ beta/(alpha)*(beta) = -1/5= -0.2.
(b) alpha^2+beta^2 =(alpha+beta)^2 -2(alpha)*(beta) = 2(-1)^2-2(5)=2-10= -8.
(c) alpha^2-beta^2 =(alpha+beta)(alpha-beta)=
(-1)√(alpha)^2+ (beta)^2 -2(alpha)(beta) =(-1)√(alpha)^2 +(beta)^2 +2(alpha)*(beta)-4(alpha)*(beta) =(-1)√(alpha+beta)^2-4(alpha)(beta) =(-1)√(-1)^2-4(5) =
(-1)√-19 = -√19i (i^2= -1) .