If the sum of the roots of the quadratic equation ax^2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then
prove that 2ca^2= bc^2 + ab^2
Answers
Step-by-step explanation:
Let α,β are roots of ax
2
+bx+c=0
Then β+α=−
a
b
;αβ=
a
c
Now
β+α= α
2
1
+
β
2
1
=
α
2
β
2
β
2
+α
2
=
α
2
β
2
(β+α)
2
−2αβ
⇒−
a
b
=
(
a
c
)
2
(−
a
b
)
2
−2
a
c
⇒−
a
b
=
(c)
2
b
2
−2ac
⇒−bc
2
=ab
2
−2ca
2
⇒2a
2
c=ab
2
+bc
2
So ab
2
,a
2
c,bc
2
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