Question

a, b, c are real numbers such that, ab + bc + ca = – 1. Prove a²
+ 5b²
+ 8c² ≥ 4.

Answer 1

We know that
(2a+5b+2c)²≥0

and also (a+6c)²≥0
Adding both equation
you will get

5a²+25b²+40c²+20ab+20bc+20ac≥0
hence

5(a²+5b²+8c²)≥-20(ab+bc+ca)

a²+5b²+8c²≥-4(-1)

a²+5b²+8c²≥4

hence proved