If
[tex] x + \frac{1}{x} = 6[/tex]
then find the value of:
1. x² + 1/x²
2.x⁴+1/x⁴
please give the answer step by step. ✔️
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If
[tex] x + \frac{1}{x} = 6[/tex]
then find the value of:
1. x² + 1/x²
2.x⁴+1/x⁴
please give the answer step by step. ✔️
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[tex]\large\underline{\sf{Given- }}[/tex]
[tex]\rm :\longmapsto\:x + \dfrac{1}{x} = 6[/tex]
[tex]\large\underline{\sf{To\:Find - }}[/tex]
[tex]\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } [/tex]
[tex]\rm :\longmapsto\: {x}^{4} + \dfrac{1}{ {x}^{4} } [/tex]
[tex]\begin{gathered}\Large{{{\underline{ \sf \: Formula \: Used - }}}} \end{gathered}[/tex]
[tex]\green{\boxed{ \bf \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy\: }}[/tex]
[tex]\large\underline{\sf{Solution-}}[/tex]
Given that,
[tex]\bf :\longmapsto\:x + \dfrac{1}{x} = 6[/tex]
On squaring both sides, we get
[tex]\rm :\longmapsto\: {\bigg(x + \dfrac{1}{x}\bigg) }^{2} = {6}^{2} [/tex]
[tex]\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 \times x \times \dfrac{1}{x} = 36[/tex]
[tex]\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 = 36[/tex]
[tex]\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } = 36 - 2[/tex]
[tex]\bf :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } = 34[/tex]
☆ On squaring both sides, we get
[tex]\rm :\longmapsto\: {\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} }\bigg) }^{2} = {(34)}^{2} [/tex]
[tex]\rm :\longmapsto\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 \times {x}^{2} \times \dfrac{1}{ {x}^{2} } = 1156[/tex]
[tex]\rm :\longmapsto\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 = 1156[/tex]
[tex]\rm :\longmapsto\: {x}^{4} + \dfrac{1}{ {x}^{4} } = 1156 - 2[/tex]
[tex]\bf :\longmapsto\: {x}^{4} + \dfrac{1}{ {x}^{4} } = 1154[/tex]
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