Question

Given that x is an integer such that
`\x√(x)-5x-9√(x)=35[`, 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:
`-(286)/(50) +/- (16i √(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√(x)-5x-9√(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