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

Question

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?

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

score 42 · Answer 1 · answered Dec 11 '14 at 21:01

42

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

answered Dec 11 '14 at 21:01

ajotatxe

65,084

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
2

Hard to resist upvoting this answer, *plus one!* – Robert Lewis Dec 11 '14 at 21:10
1

@GWAO Just think real numbers with even exponents as nonnegative numbers. – ajotatxe Dec 11 '14 at 21:40

score 5 · Answer 2 · answered Dec 11 '14 at 21:02

5

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.

answered Dec 11 '14 at 21:02

Matt Samuel

58,164

I tried that! It seemed a little too simple to be true haha – GWAO Dec 11 '14 at 21:07
1

@GWAO the lesson is that sometimes math is as simple as it looks! – Matt Samuel Dec 11 '14 at 21:08

score 4 · Answer 3 · answered Dec 11 '14 at 21:08

4

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!!!

answered Dec 11 '14 at 21:08

Robert Lewis

71,180

6

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
1

@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

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

3 Answers3