If $\sqrt{x} + \frac{1}{\sqrt{x}} = 3$ $sqrt(x) + 1/sqrt(x) = 3$ , what is $x^{2} + \frac{1}{x^{2}}$ $x^2 + 1/x^2$ ?

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If $\sqrt{x} + \frac{1}{\sqrt{x}} = 3$ $sqrt(x) + 1/sqrt(x) = 3$ , what is $x^{2} + \frac{1}{x^{2}}$ $x^2 + 1/x^2$ ?

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Answer 1 · 2018-02-12T22:24:15

$47$ $47$

Explanation:

Given:

$\sqrt{x} + \frac{1}{\sqrt{x}} = 3$ $sqrt(x)+1/sqrt(x) = 3$

Square both sides to get:

$x + 2 + \frac{1}{x} = 9$ $x+2+1/x = 9$

Subtract $2$ $2$ from both sides to get:

$x + \frac{1}{x} = 7$ $x+1/x = 7$

Square both sides to get:

$x^{2} + 2 + \frac{1}{x^{2}} = 49$ $x^2+2+1/x^2 = 49$

Subtract $2$ $2$ from both sides to get:

$x^{2} + \frac{1}{x^{2}} = 47$ $x^2+1/x^2 = 47$

Details

Note that:

${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ $(a+b)^2 = a^2+2ab+b^2$

So we find:

${(\sqrt{x} + \frac{1}{\sqrt{x}})}^{2} = {(\sqrt{x})}^{2} + 2 (\sqrt{x}) (\frac{1}{\sqrt{x}}) + {(\frac{1}{\sqrt{x}})}^{2} = x + 2 + \frac{1}{x}$ $(sqrt(x)+1/sqrt(x))^2 = (sqrt(x))^2+2(sqrt(x))(1/sqrt(x))+(1/sqrt(x))^2 = x+2+1/x$

Similarly:

${(x + \frac{1}{x})}^{2} = x^{2} + 2 (x) (\frac{1}{x}) + {(\frac{1}{x})}^{2} = x^{2} + 2 + \frac{1}{x^{2}}$ $(x+1/x)^2 = x^2+2(x)(1/x)+(1/x)^2 = x^2+2+1/x^2$

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If $\sqrt{x} + \frac{1}{\sqrt{x}} = 3$ $sqrt(x) + 1/sqrt(x) = 3$ , what is $x^{2} + \frac{1}{x^{2}}$ $x^2 + 1/x^2$ ?

1 Answer

Explanation:

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If √x+1√x=3sqrt(x) + 1/sqrt(x) = 3, what is x2+1x2x^2 + 1/x^2?

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If $\sqrt{x} + \frac{1}{\sqrt{x}} = 3$ $sqrt(x) + 1/sqrt(x) = 3$ , what is $x^{2} + \frac{1}{x^{2}}$ $x^2 + 1/x^2$ ?