Question

Given a quadratic equation say:
\[2x^{2}+8xy+7y^{2}>0\]
, how can I know that if the equation is true that it will always be more than zero without plotting its graph?

Answer 1

The squared terms are always positive so the only issue is the xy term which is also positive unless one of x or y is negative.

In this last case, if x>y then x^2 will be greater than xy or if y>x then y^2 will be greater than xy.

So the expression is always positive.

Answer 2

So there is no systematic way to find this out? What happens if the number of variables get more like 3 variables in \[3x^{2}+8xy+5xz+2yz+7y^{2}+2z^{2}>0\] I find it very difficult to look at the numbers itself because I'm worry that I may miss out any case.

Answer 3

The systematic way is to consider what is happening to the terms.

If u mean is there a formula, then no.

Answer 4

Thanks! :)