1. ## Equation

Is there any real $x$ that solves $\sqrt{x^4+x+2}=\sqrt[3]{x^5+8x+1}$?

(I checked with Wolfram Mathematica and got 2 real (and 2 complex) roots.)

Of course I only have to show that there are real root(s) and don't have to find them.

2. $f(x) = \sqrt{x^4 + x + 2} - \sqrt[3]{x^5 + 8x + 1}$

consider the signs of $f(0)$ and $f(1)$.

3. Originally Posted by skeeter
$f(x) = \sqrt{x^4 + x + 2} - \sqrt[3]{x^5 + 8x + 1}$

consider the signs of $f(0)$ and $f(1)$.
And also the sign change from $f(1)$ to $f(2)$