Question

Resolver la ecuación
$\Large{\boxed{x^{2}+ x^{-2} = 3,75}}$
​

Answer 1

${x}^{2} + {x}^{ - 2} = 3.75$

${x}^{2} + \frac{1}{ {x}^{2} } = 3.75$

$\frac{ {x}^{4} }{ {x}^{2} } + \frac{1}{ {x}^{2} } = 3.75$

${x}^{4} + 1 = 3.75$

$x = \sqrt[4]{3.75}$

X=1,4

Answer 2

${x}^{2} \: + \: {x}^{ - 2} \: = \: 3.75$

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${x}^{2} \: + \: \frac{1}{ {x}^{2} } \: = \: \frac{15}{4}$

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${x}^{2} \: + \: \frac{1}{ {x}^{2} } \: - \: \frac{15}{4} \: = \: 0$

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$\frac{4 {x}^{4} \: + \: 4 \: - \: 15 {x}^{2} }{4 {x}^{2} } \: = \: 0$

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$4 {x}^{4} \: + \: 4 \: - \: 15 {x }^{2} = \: 0$

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$4 {x}^{4} \: - \: 15 {x}^{2} \: + \: 4 \: = \: 0$

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$4 {t}^{2} \: - \: 15t \: + \: 4 \: = \: 0$

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$t \: = \: \frac{15 \: + \: \sqrt{161} }{8} \\ \\ t \: = \: \frac{15 \: - \: \sqrt{161} }{8}$

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${x}^{2} \: = \: \frac{15 \: + \: \sqrt{161} }{8} \\ \\ {x}^{2} \: = \: \frac{15 - \: \sqrt{161} }{8}$

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$x \: = \: - \frac{ \sqrt{7} \: + \: \sqrt{23} }{4}$

$x \: = \: \frac{ \sqrt{7} \: + \: \sqrt{2 3} }{4 }$

$x \: = \: - \frac{ \sqrt{7} \: - \: \sqrt{23} }{4}$

$x \: = \: \frac{ \sqrt{7} \: - \: \sqrt{23} }{4}$

Tiene 4 soluciones....

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