# 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?

## 3 Answers

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: $$\frac{54-2\sqrt{53}}{4}+\frac{26}{\sqrt{53}-1}=\frac{27-\sqrt{53}}{2}+\frac{\sqrt{53}+1}{2}=14.$$