Solve ((b-c)(1+a^2))(x+a^2)+((c-a)(1+b^2))/(x+b^2)+((a-b)(1+c^2))/(x+c^2)=0 for x

odenut6b

odenut6b

Answered question

2022-09-24

Is there any smart way to solve the equation:
( b c ) ( 1 + a 2 ) x + a 2 + ( c a ) ( 1 + b 2 ) x + b 2 + ( a b ) ( 1 + c 2 ) x + c 2 = 0

Answer & Explanation

pagellera10

pagellera10

Beginner2022-09-25Added 7 answers

I have a partial solution, as follows:
Note that ( b c ) ( 1 + a 2 ) x + a 2 = ( b c ) ( ( x + a 2 ) + ( 1 x ) ) x + a 2 = ( b c ) + ( b c ) ( 1 x ) x + a 2 . Likewise, ( c a ) ( 1 + b 2 ) x + b 2 = ( c a ) + ( c a ) ( 1 x ) x + b 2 and ( a b ) ( 1 + c 2 ) x + c 2 = ( c a ) + ( a b ) ( 1 x ) x + c 2
Now, ( b c ) ( 1 + a 2 ) x + a 2 + ( c a ) ( 1 + b 2 ) x + b 2 + ( a b ) ( 1 + c 2 ) x + c 2 = ( b c ) ( 1 x ) x + a 2 + ( c a ) ( 1 x ) x + b 2 + ( a b ) ( 1 x ) x + c 2 as ( b c ) + ( c a ) + ( a b ) = 0
Hence ( 1 x ) ( b c x + a 2 + c a x + b 2 + a b x + c 2 ) = 0 and so x=1 or b c x + a 2 + c a x + b 2 + a b x + c 2 = 0

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