TITLE: $x^4 + x^2 + 1 = 0$ has no solution in $\mathbb{R}$.
QUESTION [5 upvotes]: I need to prove that the equation $x^4 + x^2 + 1 = 0$ has no solution in $\mathbb{R}$.
How would I go about proving this?

REPLY [4 votes]: Easy!  Since $x^4 \ge 0$ and $x^2 \ge 0$ for all $x \in \Bbb R$, we have $x^4 + x^2 \ge 0$, whence $x^4 + x^2 + 1 > 0$, $\forall x \in \Bbb R$.  No zeroes in $\Bbb R$,  no solution!  QED!!!
Of course, if we allow $x \in \Bbb C$, we have a different story altogether, which is told in a different place.
Hope this helps.  Cheers!
And as ever,
Fiat Lux!!!