Question

Given that x is an integer such that mula1" title="\x\sqrt{x}-5x-9\sqrt{x}=35[" alt="\x\sqrt{x}-5x-9\sqrt{x}=35[" align="absmiddle" class="latex-formula">, find x.

Answer 1

I have done this several times and do not find that x is an integer.  I get an imaginary number.  Crazy as it seems, here's what I got all 3 times I did this:$-\frac{286}{50} +/- \frac{16i \sqrt{159} }{50}$.  That is definitely NOT an integer!

Answer 2

There appears to be a mistake in the LaTeX, so that the equation is supposed to be
$x\sqrt{x}-5x-9\sqrt{x}=35$
If we let z=√x, then this is
$z^{3}-5z^{2}-9z-35=0$

By Descartes' rule of signs, this will have one positive real root. The rational root theorem says it will be one of 1, 5, 7, or 35. It is actually z=7. The corresponding value of x is
  x = z² = 7² = 49.

A graphing calculator finds the solution to the original equation easily. (I find it useful to put it in the form f(x)=0.)

x = 49