AP EAPCET 23-Aug-2021 Shift 2 Paper

Given equation x2+px+q=0 ...(i)
On comparing with ax2+bx+c=0
a=1,b=p,c=q
Let α,β are the roots of Eq. (i)
and given that α=β2
∴α+β=

−b

a

=−p and αβ=

c

a

=q
⇒β2+β=−p and β2.β=q
and β3=q ...(ii)
and β=(q)

1

3

On taking cube both sides
(β2+β)3=(−p)3
⇒β6+β3+3β3(β2+β)=−p3
⇒q2+q+3q(−p)=−p3 [using Eq. (ii)]
⇒q2+q−3pq=−p3
⇒−p3+3pq=q2+q
⇒p(3q−p2)=q(q+1)