Question

$x = \frac{5 - \sqrt{21} }{2} \: \: find \: {x}^{2} + \frac{1}{ {x}^{2} }$
please show method​

Answer 1

Given to us :

$\sf \bullet \;\; x = \dfrac{5 - \sqrt{21}}{2}$

Need to find :

$\sf \bullet \;\; x^{2} + \dfrac{1}{x^{2}}$

Solution :-

$\sf\twoheadrightarrow \;\; x = \bigg( \dfrac{5 - \sqrt{21}}{2} \bigg)$

$\sf\twoheadrightarrow \;\; \dfrac{1}{ x} = \bigg( \dfrac{2}{5 - \sqrt{21}} \bigg)$

~By rationalizing the denominator

$\sf\twoheadrightarrow \;\; \dfrac{1}{ x} = \bigg( \dfrac{2}{5 - \sqrt{21}} \bigg) \times \dfrac{5 + \sqrt{21}}{5 + \sqrt{21}}$

$\sf\twoheadrightarrow \;\; \dfrac{1}{ x} = \bigg( \dfrac{2( 5 + \sqrt{21})}{(5 - \sqrt{21})(5 + \sqrt{21})} \bigg)$

$\sf\twoheadrightarrow \;\; \dfrac{1}{ x} = \bigg( \dfrac{2( 5 + \sqrt{21})}{(5)^{2} - (\sqrt{21})^{2}} \bigg)$

$\sf\twoheadrightarrow \;\; \dfrac{1}{ x} = \bigg( \dfrac{2( 5 + \sqrt{21})}{25 - 21 } \bigg)$

$\sf\twoheadrightarrow \;\; \dfrac{1}{ x} = \bigg( \dfrac{2( 5 + \sqrt{21})}{4 } \bigg)$

$\sf\twoheadrightarrow \;\; \dfrac{1}{ x} = \bigg( \dfrac{5 + \sqrt{21}}{2} \bigg)$

$\sf\twoheadrightarrow \;\; x + \dfrac{1}{ x} = \bigg( \dfrac{5 + \sqrt{21}}{2} \bigg) + \bigg( \dfrac{5 - \sqrt{21}}{2} \bigg)$

$\sf\twoheadrightarrow \;\; x + \dfrac{1}{ x} = \bigg( \dfrac{5 + \sqrt{21} + 5 - \sqrt{21}}{2} \bigg)$

$\sf\twoheadrightarrow \;\; x + \dfrac{1}{ x} = \bigg( \dfrac{5 + 5}{2} \bigg)$

$\sf\twoheadrightarrow \;\; x + \dfrac{1}{ x} = \bigg( \dfrac{10}{2} \bigg)$

$\sf\twoheadrightarrow \;\; x + \dfrac{1}{ x} = 5$

~Squaring on both the sides of this equation.

$\sf\twoheadrightarrow \;\; \bigg( x + \dfrac{1}{ x} \bigg)^{2} = 5^{2}$

$\sf\twoheadrightarrow \;\; x^{2} + \bigg( 2 \times x \times \dfrac{1}{x} \bigg) + \dfrac{1}{x^{2}} = 5^{2}$

$\sf\twoheadrightarrow \;\; x^{2} + \bigg( \dfrac{2x}{x} \bigg) + \dfrac{1}{x^{2}} = 5^{2}$

$\sf\twoheadrightarrow \;\; x^{2} +2+ \dfrac{1}{x^{2}} = 5^{2}$

$\sf\twoheadrightarrow \;\; x^{2} + \dfrac{1}{x^{2}} = 5^{2} - 2$

$\sf\twoheadrightarrow \;\; x^{2} + \dfrac{1}{x^{2}} = 25 - 2$

$\boxed{\bf{ x^{2} + \dfrac{1}{x^{2}} = 23 }} \;\; \bigstar$

    _________

• Henceforth, the value is 23.