Question

Prove that a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)+3=0 when a+b+c=0.

Answer 1

Step-by-step explanation:

→ a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)+3=0

→ a(b+c)/bc + b(a+c)/ac + c(a+b)/ab + 3=0

→ a²(b+c)/abc + b²(a+c)/abc + c²(a+b)/abc + 3=0

→ { a²(b+c)+b²(a+c)+c²(a+b)+3abc }/abc=0

→ a²(b+c)+b²(a+c)+c²(a+b)+3abc=0

→ a²b+a²c+b²a+b²c+c²a+c²b+3abc = 0

→ a²b+b²a+abc+a²c+c²a+abc+c²b+b²c+abc=0

→ ab(a+b+c)+ac(a+b+c)+bc(a+b+c)=0

→ (a+b+c)(ab+ac+bc)=0

→ a+b+c=0. proved