AP EAPCET 19 May 2024 Shift 2 Solved Paper

To find α−1+β−1+γ−1, we consider that α,β, and γ are the roots of the equation:
x3+ax2+bx+c=0.
From Vieta's formulas, we know:
‌α+β+γ=−a,
‌αβ+βγ+γα=b,
‌αβγ=−c.
To find the sum of the reciprocals of the roots:
α−1+β−1+γ−1=‌

1

α

+‌

1

β

+‌

1

γ

Utilizing the identity for sums of reciprocals, this can be rewritten as:
α−1+β−1+γ−1=‌

βγ+αγ+αβ

αβγ

.
Substituting the known values from Vieta's formulas, we get:
α−1+β−1+γ−1=‌

b

−c

=−‌

b

c

.