if a,b,c are all non zero and a+b+c=0, prove that a^2/bc+b^2/ca+c^2/ab=3

Question

if a,b,c are all non zero and a+b+c=0, prove that a^2/bc+b^2/ca+c^2/ab=3

if a,b,c are non zero,

a+b+c=0

prove that

 

Answer 1

We start this proof by letting a, b, c be non zero real numbers and we assume a+b+c=0. This implies that a = -(b+c), b = -(a+c), c = -(a+b). Plugging these into the expression we need to work with we get:

(b+c)²/(bc) + (a+c)²/(ac) + (a+b)²/(ab). Notice I did not include any of the negatives the numerators, since they were being raised to the 2nd power the negatives are canceled.

Next we FOIL the numerators and do some canceling out then you should get: (b/c) + 2 + (c/b) + (a/c) + 2 + (c/a) + (a/b) + 2 +(b/a).

Simplifying this expression further by combining like terms we get: 6 + (c+b)/a + (a+c)/b + (a+b)/c.

From our assumption we get: 6 + (-a/a) + (-b/b) + (-c/c) = 6 - 1 - 1 -1 = 3.