Question

if x=a+b, y=b+c, z=c+a then find the value of x^2+y^2+z^2-xy-yz-zx Pls answer it with detailed steps and correctly.

Answer 1

x = a + b

y = b + c

z = c + a

x² + y² + z² – xy – yz – zx

= ( a + b )² + ( b + c )² + ( c + a )² – ( a + b ) ( b + c ) – ( b + c ) ( c + a ) – ( c + a ) ( a + b )

= a² + b² + 2ab + b² +c² + 2bc + c² + a² – ( a + b ) ( b + c ) – ( b + c ) ( c + a ) – ( c + a ) ( a + b )

= 2a² + 2b² + 2c² + 2ab + 2bc + 2ca – [ ab + ac + b² + bc + bc + ab + c² + ca + ca + bc + a² + ab ]

= 2a² – 2b² – 2c² + 2ab + 2bc + 2ca – [ 3ab + 3bc + 3ca + a² + b² + c² ]

= 2a² – a² + 2b² – b² + 2c² – c² + 2ab – 3ab + 2bc – 3bc + 2ca – 3ca

= a² + b² + c² – ab – bc – ca