text: # Solve without a calculator: If $x+\sqrt{x}=13$ then $x+\frac{13}{\sqrt{x}}=?$
  
  $$x+\sqrt{x}=13$$ $$x+\frac{13}{\sqrt{x}}=?$$
  
  I tried to square(and also triple in another attempt) both sides of both of the equations hoping that I would find some expression to plug in. It didn't help.
  
  Then, I tried to simply bring the second equation to a common denominator.It didn't help neither.
  
  Then I found the $$x$$ value ($$9.86$$) from the first equation by using a calculator and then plugged in to the second equation and got this expression $$\frac{493}{50}+65\sqrt{\frac{2}{493}}$$
  
  Now, Is this question solvable without a calculator?
  
  Note that $$x+\frac{13}{\sqrt{x}} = x + \frac{x+\sqrt{x}}{\sqrt{x}} = x+\sqrt{x}+1=14.$$
  
  Divide both sides of your equation by the square root of $$x$$.
  
  $$x$$/$$\sqrt{x}$$ + $$\sqrt{x}$$/$$\sqrt{x}$$ = 13/$$\sqrt{x}$$
  
  or
  
  $$\sqrt{x}$$ + 1 = 13/$$\sqrt{x}$$
  
  Now substitute for $$\sqrt{x}$$ from the original equation, i.e. $$\sqrt{x}$$ = 13 - $$x$$
  
  13 - $$x$$ + 1 = 13/$$\sqrt{x}$$
  
  14 = $$x$$ + 13/$$\sqrt{x}$$
  
  Alternatively, solve the first equation: $$\sqrt{x}=\frac{-1+\sqrt{53}}{2}$$ and sub in the second equation: $$\frac{54-2\sqrt{53}}{4}+\frac{26}{\sqrt{53}-1}=\frac{27-\sqrt{53}}{2}+\frac{\sqrt{53}+1}{2}=14.$$
  
  If $$x + \frac{13}{\sqrt x} = a$$, then:
  
  $$13a = (x+\sqrt x)(x + \frac{13}{\sqrt x})$$ $$= x^2 + 13\sqrt x + x\sqrt x + 13$$
  
  Now use the equality $$x + \sqrt x = 13$$:
  
  $$13a = x^2 + (x + \sqrt x)\sqrt x + x \sqrt x + (x +\sqrt x)$$ $$(x + \sqrt x)a = x^2 + x\sqrt x + x + x \sqrt x + x + \sqrt x$$ $$(x + \sqrt x)a = (x + \sqrt x)x + \sqrt x(x + \sqrt x) + (x + \sqrt x)$$ $$a = x + \sqrt x + 1$$ $$a = 13 + 1 = 14$$
  
  You can get to the answer much quicker using Math Lover's answer however.