Question

$if \: x + \frac{1}{x} = 3 \: \: \\ then \: the \: value \: of \: x - \frac{1}{x} \: is$
$(a) \: + \sqrt{5} \\ (b) - \sqrt{5} \\ (c) + or - \: \sqrt{5}$

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Answer 1

Answer:

$x + \frac{1}{x} = 3$

${(x + \frac{1}{x}) }^{2} = {3}^{2}$

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

${x}^{2} + \frac{1}{ {x}^{2} } = 9 - 2$

${x}^{2} + \frac{1}{ {x}^{2} } = 7$

Now,

${(x - \frac{1}{x}) } ^{2} + 2 = 7$

${(x - \frac{1}{x}) }^{2} = 7 - 2$

${(x - \frac{1}{x}) }^{2} = 5$

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

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

Answer is this +-√5 as whether the number is -√5 or√5 it's square will be 5