x-2-1-x-2-51-find-x-

Question Number 83672 by jagoll last updated on 05/Mar/20

x^{2} + \frac{1}{x^{2}} = 51

f i n d x

Answered by john santu last updated on 05/Mar/20

x^{2} + \frac{1}{x^{2}} = 51 \Leftrightarrow {(x + \frac{1}{x})}^{2} - 2 = x^{2} + \frac{1}{x^{2}}

{(x + \frac{1}{x})}^{2} = 53 \Rightarrow x + \frac{1}{x} = \pm \sqrt{53}

case (1) \Rightarrow x + \frac{1}{x} = \sqrt{53}

x^{2} - x \sqrt{53} + 1 = 0

x = \frac{\sqrt{53} \pm 7}{2} = {\begin{cases} x = \frac{\sqrt{53} + 7}{2} \\ x = \frac{\sqrt{53} - 7}{2} \end{cases}

case (2) \Rightarrow x + \frac{1}{x} = - \sqrt{53}

x^{2} + x \sqrt{53} + 1 = 0

x = \frac{- \sqrt{53} \pm 7}{2} = {\begin{cases} x = \frac{7 - \sqrt{53}}{2} \\ x = \frac{- 7 - \sqrt{53}}{2} \end{cases}

Answered by mr W last updated on 05/Mar/20

{(x + \frac{1}{x})}^{2} = 53 \Rightarrow x + \frac{1}{x} = \pm \sqrt{53}

{(x - \frac{1}{x})}^{2} = 49 \Rightarrow x - \frac{1}{x} = \pm 7

\Rightarrow x = \frac{1}{2} (\pm \sqrt{53} \pm 7) = {\begin{cases} \frac{\sqrt{53} + 7}{2} \\ \frac{\sqrt{53} - 7}{2} \\ \frac{- \sqrt{53} + 7}{2} \\ \frac{- \sqrt{53} - 7}{2} \end{cases}

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