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

Question

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

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Answer 1 · 2017-10-21T14:21:30

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 }#

#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 · 2017-10-21T15:45:20

#x = 9#

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#