How to prove a function has no real solutions So I have the equation:
$$
0.7=0.04(10+x)+\frac{2100}{(10+x)^3}
$$

And I have tried solving for x and have not succeeded, I believe it only has complex solutions.

So my question is, how do I go about actually proving this equation has no solutions for $x \ \varepsilon\ R$

Thank you for the help!
You can define the function $f(x)=0.04(10+x)+\frac{2100}{(10+x)^3}-0.7$.
If the equation has a real solution, then this function intersects the x axis at some point. It's a pretty "nice" function (continuous and differentiable at almost any point) which means you can use its derivatives and limits at specific points in order to practically know anything about its graph. Mainly - whether or not it intersects the x axis.
Multiplying by $$100(10+x)^3$$ and simplifying we get

$$2\,{x}^{4}+45\,{x}^{3}+150\,{x}^{2}-2500\,x+90000=0$$
This equation has only complex Solutions.