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If $x^3=a+1$ and $x+(b/x)=a$ Then $x$ equals?? Please help in solving these equations. I can't get it how to solve $1$ variable by two equations

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  • $\begingroup$ Are $a$ and $b$ supposed to be unknowns or parameters? Are they (and $x$) real, integer, natural numbers? $\endgroup$ – Arnaud D. Oct 16 '16 at 12:57
  • $\begingroup$ Sorryt I forgot to tell.x is the variable.a and b are constant natural numbers please keep the answer in terms of a and b $\endgroup$ – SayDude Oct 16 '16 at 13:03
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Notice that $x+\dfrac{b}{x}=a$ is equivalent to $x^2=ax-b$. Now multiplying this by $x$ yields $$a+1=x^3=ax^2-bx=a(ax-b)-bx=(a^2-b)x-ab,$$ and thus $x=\dfrac{1+a+ab}{a^2-b}$. Note that there must be some conditions on $a$ and $b$ for this to hold!

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You have actually three unknowns and only two equation so the value of $x$ is not unique. One has for example, the integer solutions (which is not necessary) $$(x,a,b)=(2,7,35),(1,0,-1),(3,26,69)$$ and infinitely many more (take a value of $a$ such that $a+1$ be a cube so you have a value for $x$ and a value for $b$).

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  • $\begingroup$ You are welcome. $\endgroup$ – Piquito Oct 16 '16 at 15:47

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