# $x^4 + x^2 + 1 = 0$ has no solution in $\mathbb{R}$. [closed]

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?

## closed as off-topic by Watson, Saad, Brahadeesh, Arnaud D., I am BackNov 28 '18 at 11:46

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• Have you tried factoring the left-hand side? – Nick Dec 11 '14 at 21:07

$x^2$ and $x^4$ are not negative, so $$1+x^2+x^4\ge1$$ for any real number $x$.

• This is better than my answer. On the bright side, at least my answer will work. – Matt Samuel Dec 11 '14 at 21:03
• This is the type of proof I was looking for! It seems obvious after reading your answer, but how would you come up with that in the first place? – GWAO Dec 11 '14 at 21:07
• Hard to resist upvoting this answer, plus one! – Robert Lewis Dec 11 '14 at 21:10
• @GWAO Just think real numbers with even exponents as nonnegative numbers. – ajotatxe Dec 11 '14 at 21:40

I would solve the equation. Take $y=x^2$. Then the equation is $y^2+y+1=0$. This is quadratic and easy to solve.

• I tried that! It seemed a little too simple to be true haha – GWAO Dec 11 '14 at 21:07
• @GWAO the lesson is that sometimes math is as simple as it looks! – Matt Samuel Dec 11 '14 at 21:08

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!!!

• Do you think you could use a few more exclamation points in there? ;) – augurar Dec 11 '14 at 21:56
• No such thing as too many exclamations. Helps get your point across. (!!!) – Dasherman Dec 11 '14 at 22:02
• @Dasherman: I couldn't agree more !!!!!!!!!!!!!!!!!!!!! – Robert Lewis Dec 11 '14 at 22:28
• @augurar: try 'em, you'll like 'em . . . Hell! You'll Love 'em !!!!!!!!!!!!!!!!!!!!! – Robert Lewis Dec 11 '14 at 22:29
• And a Jovial Yule Season to All and Sundry !!!!!!!!!!!!!!!!!!!!! Ho ho ho!!!!!!!!!!!!!!!!!!!!! – Robert Lewis Dec 11 '14 at 22:29