The third expression

Algebra Level 3

a , b , c a,b,c are 3 real numbers such that a ≠ b ≠ c a\neq b\neq c and a 2 ( b + c ) = b 2 ( c + a ) = 2017 { a }^{ 2 }\left( b+c \right) ={ b }^{ 2 }\left( c+a \right) =2017 .

What is the value of M = c 2 ( a + b ) M={ c }^{ 2 }\left( a+b \right) ?


The answer is 2017.

Linkin Duck by Linkin Duck , 33, Vietnam

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1 solution

Linkin Duck
Mar 28, 2017

From a 2 ( b + c ) = b 2 ( c + a ) { a }^{ 2 }\left( b+c \right) ={ b }^{ 2 }\left( c+a \right)

⟹ ( a − b ) ( a b + b c + c a ) = 0 ⟹ a b + b c + c a = 0 ( a ≠ b ) ⟹ ( b − c ) ( a b + b c + c a ) = 0 ⟹ b 2 a + b 2 c − b c 2 − a c 2 = 0 \Longrightarrow \left( a-b \right) \left( ab+bc+ca \right) =0\\ \Longrightarrow ab+bc+ca=0\quad \left( a\neq b \right) \\ \Longrightarrow \left( b-c \right) \left( ab+bc+ca \right) =0\\ \Longrightarrow { b }^{ 2 }a+{ b }^{ 2 }c-b{ c }^{ 2 }-a{ c }^{ 2 }=0

So, M = c 2 ( a + b ) = b 2 ( c + a ) = 2017. M={ c }^{ 2 }\left( a+b \right) ={ b }^{ 2 }\left( c+a \right) =2017.

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