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I have a problem from my textbook. By using $x - 3\sqrt{\frac{5}{x}} = 8$ how can we find the value of $x - \sqrt{5x}$. I have derived the equation that's given so much, but i couldn't find the answer. Solvings or hints are appreciated.

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    can you solve the equation $$x-3\sqrt{\frac{5}{x}}=8$$? – 2017-02-20
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    If i say $\sqrt{x} = a$ i will get $a^3-8a-3\sqrt{5} = 0$. Looks not too simple. – 2017-02-20

3 Answers 3

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The function $f(x)=x-3\sqrt{5\over x}$ increases from $-\infty$ at $x=0$ to $+\infty$ as $x\to\infty$, so the equation $x-3\sqrt{5\over x}=8$ has exactly one solution for $x\gt0$. Multiplying through by $x$ and moving everything to the left hand side, we have

$$x^2-8x-3\sqrt{5x}=0$$

Now let $u=x-\sqrt{5x}$. We can rewrite this equation as

$$x^2-11x+3u=0$$

On the other hand, moving the $-3\sqrt{5x}$ to the right hand side and squaring gives $x^4-16x^3+64x^2=45x$, or

$$(x^3-16x^2+64x-45)x=(x-5)(x^2-11x+9)x=0$$

But $x=0$ is obviously not a solution to $x-3\sqrt{5\over x}=8$. Nor is $x=5$. So the (unique positive) solution must satisfy $x^2-11x+9=0$. But since it also satisfies $x^2-11x+3u=0$, we see that $9=3u$, or $u=3$. We thus find that $x-\sqrt{5x}=3$.

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    That's the approach i've expect. Thanks so much! – 2017-02-20
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we can write $$x-8=3\sqrt{\frac{5}{x}}$$ after squaring this equation we get $$x^2-16x+64=9\cdot \frac{5}{x}$$ multiplying by $$x\ne 0$$ we obtain $$x^3-16x^2+64x-45=0$$ factorizing this equation we get $$(x-5)(x^2-11x+9)=0$$ can you finish now?

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    As i saw from the Hat's answer, there is a root that's $\frac{11+\sqrt{85}}{2}$(can be found from $x^2−11x+9$ at your answer). But i suppose there should be a simpler way to solve this problem. A simple derivation maybe. – 2017-02-20
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$x - 3 \sqrt{\frac{5}{x}} = 8$

$\rightarrow$ $(x-3 \sqrt\frac{5}{x})^2 = 64$

$\rightarrow$ $x^2 - 6\sqrt{5x} + \frac{45}{x} = 64$

$\rightarrow$ $6x\sqrt{5x} = -64x + x^3 + 45$

$\rightarrow 180x^3 = x^6 -128x^4 +90x^3 +4096x^2 -5760x + 2025$

$\rightarrow$ $x^6 - 128x^4 - 90x^3 + 4096x^2 - 5760x + 2025 = 0$

Graphically (or through lots of algebra), we see that $x = \frac{1}{2}(11 + \sqrt85)$ is a solution to this equation

So, $x - \sqrt{5x} = \frac{1}{2}(11 + \sqrt{85}) - \sqrt{5 \cdot \frac{1}{2}(11 + \sqrt{85}}) = 3$

The answer is 3.