Question

If alpha and beta are zeroes of polynomial ax^2+bx+c then,evaluate
1)1 by alpha+1 by beta-2alphabeta

Answer 1

Given polynomial,
ax^2+bx+c=0
sum of the zeroes=alpha+beta=-b/a......1
product of the zeroes=alpha×beta=c/a...2
=1/alpha+1/beta-2alphabeta
=(alpha+beta)/(alphabet)-2alphabeta
=(-b/a)/c/a-2×c/a
=-b/a×a/c-2c/a
=-b/c-2c/a
=-(ba+2c^2)/ac
this is the required value.

Answer 2

alpha and beta are the roots of
ax² + bx + c =0

alpha + beta = -b/a
alpha.beta = c/a

now,
1/alpha + 1/beta -2alpha.beta

=(beta + alpha )/alpha .beta -2alpha.beta

= ( -b/a)/(c /a) -2c/a

=-b/c -2c/a

= -( ab +2c²)/ac