Completing the square for a quartic expression

Question

By completing the square, find (for real $x$) the minimum value of: $$x^4 + 2x^2 + 2.$$

Answer 1

$$x^4+2x^2+2=(x^2+1)^2+1$$

Now for real $x,x^2\ge0$

Answer 2

We can observe that $$x^4+2x^2\geq 0$$ for all $x\in\mathbb R$ and so, $$x^4+2x^2+2\geq 2$$ for all $x\in \mathbb R$. Moreover, for $x=0$, we have that $$0^4+2\cdot 0^2+2=2$$ and so $2$ is the minimum.

Answer 3

$$ x^4+x^2+2=(x^2+1)^2+1\\x^2\geq 0\\x^2+1 \geq 0+1 \\(x^2+1)^2 \geq (1)^2\\so\\(x^2+1)^2+1\geq 1+1 $$