MHB Equation 2: Prove that ##x^2+2x\sqrt x+3x+2\sqrt x+1=0##

[solakis1]

I checked the following equation with Wolfram\Alpha and the answer was no real solution
How can we prove that?
$x^2+2x\sqrt x+3x+2\sqrt x+1=0$

 

[topsquark]

I checked the following equation with Wolfram\Alpha and the answer was no real solution
How can we prove that?
$x^2+2x\sqrt x+3x+2\sqrt x+1=0$

You pretty much are stuck with two choices: Ferrari's method or graphing. I'd choose graphing!

-Dan

 

[Opalg]

Spoiler

$f(x) = x^2 + 2x\sqrt x + 3x + 2\sqrt x + 1 = (x + \sqrt x + 1)^2$, so if $f(x) = 0$ then $x + \sqrt x + 1 = 0$. That is a quadratic equation for $\sqrt x$, with solutions $\sqrt x = \frac12(-1 \pm i\sqrt3)$. If $\sqrt x$ is non-real then so is $x$. Therefore the equation $f(x) = 0$ has no real solutions.

 

[topsquark]

[sp]Well, if $\sqrt{x}$ is not pure imaginary then x is complex, anyway.[/sp]

Nice catch!

-Dan

 

[solakis1]

ha, ha so easy solution