Question

If a,b,c are all non- zero number and a+b+c =0 .Prove that a²/bc+ b²/ca+ c²/ab =3

Answer 1

use the formula a^3 + b^3 + c^3 = 3abc  

then divide lhs and rhs by abc 

Answer 2

Answer

Let's simplify the given equation first:

$\frac{ {a}^{2} }{bc} + \frac{ {b}^{2} }{ca} + \frac{ {c}^{2} }{ab} \\ \\ = > \frac{ {a}^{3} + {b}^{3} + {c}^{3} }{abc}$

We know an identity:

When: a + b + c = 0;

${a}^{3} + {b}^{3} + {c}^{3} = 3abc$

Applying the same here:

$\frac{ {a}^{3} + {b}^{3} + {c}^{3} }{abc} \\ \\ = > \frac{3abc}{abc} \\ \\ = > 3$