Solve without a calculator: If $x+\sqrt{x}=13$ then $x+\frac{13}{\sqrt{x}}=?$

Question

$$x+\sqrt{x}=13$$ $$x+\frac{13}{\sqrt{x}}=?$$

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?

Answer 1

Note that $$x+\frac{13}{\sqrt{x}} = x + \frac{x+\sqrt{x}}{\sqrt{x}} = x+\sqrt{x}+1=14.$$

Answer 2

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/$\sqrt{x}$

Answer 3

Alternatively, solve the first equation: $$\sqrt{x}=\frac{-1+\sqrt{53}}{2}$$ and sub in the second equation: $$\frac{54-2\sqrt{53}}{4}+\frac{26}{\sqrt{53}-1}=\frac{27-\sqrt{53}}{2}+\frac{\sqrt{53}+1}{2}=14.$$

Answer 4

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.