# 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.
The text passage contains a mathematical question and provides a detailed solution to it, including multiple methods to solve the problem.
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Given a text passage, determine:
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First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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# 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.
user
The text passage contains a mathematical question and provides a detailed solution to it, including multiple methods to solve the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER