Question

How many positive real number X Satisfy the given equation:?

`x^3-3 mod x+2=0`

Answer 1

The only positive integer solution is `x=9`

If we allow non-integer moduli, then there are additional solutions:

`x in { 7/2, 5/3, 3/4, 1/5 }`

Explanation:

`x^3-3 = x^3+2x^2-2x^2-4x+4x+8-11`

`color(white)(x^3-3) = (x+2)(x^2-2x+4)-11`

So we require:

`-11 -= 0` mod `x+2`

So the only positive integer solution is `x=9`

If we allow non-integer moduli then there are additional positive rational solutions of the form:

`x = 11/n-2`

namely:

`x = 11/2-2 = 7/2`

`x = 11/3-2 = 5/3`

`x = 11/4-2 = 3/4`

`x = 11/5-2 = 1/5`

Answer 2

`x = 9`

Explanation:

Solving

`x^3 - 3= (x^2 + a x + b) (x + 2) + c`

we have

`a=-2, b=4, c=-11`

now making

`k(x+2) = c` we have `x=-(11 + 2 k)/k`

As long as `x` needs to be a positive integer we will choose `k` such that

`-(11 + 2 k)/k = n` or `k = -11/(2 + n)` with `{k,n} in ZZ`

and then the only choice is for `n = x=9` when `k = -1`