Question

if x=9-4√5 then find √x-1/√x

Answer 1

Hi...☺

Here is your answer...✌

$x = 9 - 4 \sqrt{5}$

$\sqrt{x} = \sqrt{9 - 4 \sqrt{5} } \\ \\ \sqrt{x} = \sqrt{ 5 + 4 - 4 \sqrt{5} } \\ \\ \sqrt{x} = \sqrt{ {(\sqrt{5})}^{2} + {2}^{2} - 2 \times \sqrt{5} \times 2 } \\ \\ \sqrt{x} = \sqrt{{ ( \sqrt{5} - 2) }^{2} } \\ \\ \sqrt{x} = \sqrt{5} - 2$

And

$\frac{1}{ \sqrt{x} } = \frac{1}{ \sqrt{5} - 2 } \\ \\ \frac{1}{ \sqrt{x} } = \frac{1}{ \sqrt{5} - 2} \times \frac{ \sqrt{5} + 2 }{ \sqrt{5} + 2} \\ \\ \frac{1}{ \sqrt{x} } = \frac{ \sqrt{5} + 2 }{{( \sqrt{5})}^{2} - {2}^{2} } \\ \\ \frac{1}{ \sqrt{x} } = \frac{ \sqrt{5} + 2 }{5 - 4} = \frac{ \sqrt{5} + 2}{1} \\ \\ \frac{1}{ \sqrt{x} } = \sqrt{5} + 2$

Now,

$\sqrt{x} - \frac{1}{\sqrt{x} } \\ \\ = \sqrt{5} - 2 - \sqrt{5} - 2 \\ \\ = -4$

Answer 2

Answer:

-4

Step-by-step explanation:

We are given x = 9 - 4√5

We can write,

x = 5 + 4 - 4√5  

x=(√5)² + 2² - 2(2)(√5)

x=(√5 - 2)²

Now,

√x = √5 - 2

For finding 1/√x, we do rationalization:

$\frac{1}{\sqrt{x} } = \frac{1}{\sqrt{5} -2} \\\\\frac{1}{\sqrt{x} } = \frac{1}{\sqrt{5} -2} *\frac{\sqrt{5} +2}{\sqrt{5} +2} \\\\\frac{1}{\sqrt{x} } ={\sqrt{5} +2$

Now, to find:

$\sqrt{x} -\frac{1}{\sqrt{x} } = \sqrt{5} -2 -( \sqrt{5} +2)\\\\\sqrt{x} -\frac{1}{\sqrt{x} } = \sqrt{5} -2 -\sqrt{5} -2\\\\ \sqrt{x} -\frac{1}{\sqrt{x} } = -4$

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