<s>[INST] 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?
 [/INST] 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.
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