If α and β are the roots of ax^2 + 2bx + c = 0 where s (α+δ) and (β+δ) are the roots of Ax^2 + 2Bx + C = 0 for some constant δ, then prove that b^2 - ac / B^2 - AC = (a/A)^2 .
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Answer:
Let α,β be roots of ax
3
+bx+c=0
γ,δ be roots of px
2
+2qx+r=0
β
α
=
δ
γ
…………(1)
and
α
β
=
γ
δ
………….(2)
(1)+(2)
⇒
β
α
+
α
β
=
δ
γ
+
γ
δ
=
αβ
2
2
+β
2
+2=
γδ
γ
2
+δ
2
+2
⇒
αβ
(α)
2
+(β)
2
+2αβ
=
γδ
γ
2
+δ
2
+2γδ
=
αβ
(α+β)
2
=
γδ
(γ+δ)
2
⇒
a
c
a
2
4b
2
=
p
r
p
2
4q
2
⇒
ac
b
2
=
p
r
q
2
.
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