Question :

If the roots of the equation ( a − b ) ² + ( b − c ) ² + ( c − a )² = 0 are equal, prove that 2 a = b + c .
Submitted on 8/3/2024 | Answered by Vandana Rana

Answer :


Given,
(
a

b
)
x2

+
(
b

c
)
x
+
(
c

a
)
=
0

are equal.


Then the discriminant (D)
=
0


Then
(
b

c
)2


4
(
c

a
)
(
a

b
)
=
0



b2


2
b
c
+
c2



4
(
a
c

b
c

a2

+
a
b
)
=
0



b2


  • 2
    b
    c
    +
    c2



    4ac + 4bc + 4a2 - 4
    ab=0



    b2 + c2 + 4a2 + 2bc - 4ac - 4ab =0


    (
    b
    +
    c

    2
    a
    )2

    =
    0


    2
    a
    =
    b
    +
    c