Question

if x=√5-2÷√5+2, find the value of: x^2+1/x^2​

Answer 1

Answer:

The Answer is 18.

Step-by-step explanation:

if x=√5-2÷√5+2, find the value of: x^2+1/x^2

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Answer 2

Step-by-step explanation:

$\mathrm{ \huge{\underline{Question : }}}$

$\mathrm{If \: x = \frac{ \sqrt{5} - 2}{ \sqrt{5} + 2} ,find \: the \: value \: of : {x}^{2} + \frac{1}{ {x}^{2} } }.$

$\mathrm{ \underline {\huge{To \: find \: out:}}}$

$\mathrm{The \: value \: of : {x}^{2} + \frac{1}{ {x}^{2} } } =$

$\mathrm{ \huge \underline{Solution : }}$

Let's solve the problem

we have,

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

$\mathrm{∴ \: \: \: \frac{1}{x} = \frac{ \sqrt{5} + 2}{ \sqrt{5} - 2 } }$

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

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

$⟹ \mathrm{x + \frac{1}{x} } = \frac{5 + 4 - 4 \sqrt{5} + 5 + 4 + 4 \sqrt{5} }{5 - 4}$

$⟹ \mathrm{x + \frac{1}{x} } = \frac{5 + 4 - 4 \sqrt{5} + 5 + 4 + 4 \sqrt{5} }{1}$

$⟹ \mathrm{x + \frac{1}{x} } = 5 + 4 - \cancel{4 \sqrt{5} } + 5 + 4 + \cancel{4 \sqrt{5} }$

$⟹ \mathrm{x + \frac{1}{x} } = 5 + 4 + 5 + 4$

$⟹ \mathrm{x + \frac{1}{x} } = 18$

Squaring on both sides using algebraic Identity (a+b)² = a²+2ab+b² we get

$\mathrm { \bigg(x + \frac{1}{x} \bigg)^{2} } = (18 {)}^{2}$

$⟹ \mathrm{ {x}^{2} + 2( \cancel{x}) \bigg( \frac{1}{ \cancel{x}} \bigg) + \bigg( \frac{1}{x} \bigg)^{2} } = 324$

$⟹ \mathrm{ {x}^{2} + 2 + \frac{1}{ {x}^{2} } } = 324$

$⟹ \mathrm{ {x}^{2} + \frac{1}{ {x}^{2} } } = 324 - 2$

$⟹ \mathrm{ {x}^{2} + \frac{1}{ {x}^{2} } } = 322$

$\mathrm{ \underline{Answer : } \: Hence, the \: value \: of \: {x}^{2} + \frac{1}{ {x}^{2} } \: is \: 322. }$

Used formulae:—

• (a+b)² = a²+2ab+b²

• (a-b)² = a² -2ab +b²

• (a-b)(a+b) = a² - b²

• (a+b)(a-b) = a² - b²

Siddhi :)