Question

why a^2+ab+b^2≥0 ؟

Answer 1

if a and b have the same sign (either both plus or both negative)
then the + a b term is positive. Also, both a^2 and b^2 is positive
so in this case the sum >=0 (0 only if both a and b equal 0)

Answer 2

on the other hand, we can write
a^2+ab+b^2
as
a^2+ab+b^2 + ab - ab (note: we are adding "zero" in the form ab-ab )
which is
a^2 +2ab + b^2 - ab
(a+b)^2 -ab

this is another way to write the original expression (which we know is >=0 if a and b have the same sign)

However, this alternate expression has the term -ab
which will be positive if a and b have *opposite* signs
and as (a+b)^2 is positive, the whole expression is positive.

Answer 3

Also,
if either a or b (or both is zero) then a^2 +ab+b^2 >=0

thus we have shown that the expression is >= 0 for the cases of
1) a and b have the same sign
2) a and b having "opposite" signs
3) either a, b or both is zero

This covers all possibilities, and we conclude the expression >= 0

Answer 4

thank you so much :)