How to prove the below inequality is always satisfied?

Question

Given inequality -: x² + 2xy + 2y² > 0 always satisfies the inequality for every x and y, where x and y are real numbers(x and y are not equal to 0).

How to prove this without using any graph? Is there any mathematical formula or something like that?

Second example -: x² + 10xy + 2y² > 0 does not satisfy the inequality always.

Yes, "something like that". You can really prove it mathematically. See also the example of $x^2+xy+y^2\ge 0$. — Dietrich Burde, Sep 08 '19 at 15:49
To prove the second example doesn’t satisfy the inequality always, you just need a counterexample, such as $x=1, y=-1$ — J. W. Tanner, Sep 08 '19 at 16:32

score 1 · Answer 1 · answered Sep 08 '19 at 15:48

1

It is $$x^2+2xy+y^2+y^2=(x+y)^2+y^2>0$$

answered Sep 08 '19 at 15:48

1 Answers1