Question

if x+1/x=2,find X2+1/X2 and x4+1/x4​

Answer 1

EXPLANATION.

⇒ x + 1/x = 2.

As we know that,

Squaring on both sides of the equation, we get.

⇒ (x + 1/x)² = (2)².

⇒ (x² + 1/x² + 2(x)(1/x)) = 4.

⇒ x² + 1/x² + 2 = 4.

⇒ x² + 1/x² = 4 - 2.

⇒ x² + 1/x² = 2.

Again squaring on both sides of the equation, we get.

⇒ [x² + 1/x²]² = (2)².

⇒ [x⁴ + 1/x⁴ + 2(x²)(1/x²)] = 4.

⇒ [x⁴ + 1/x⁴ + 2] = 4.

⇒ x⁴ + 1/x⁴ = 4 - 2.

⇒ x⁴ + 1/x⁴ = 2.

Value of :

x² + 1/x² = 2   &   x⁴ + 1/x⁴ = 2.

Answer 2

$\large\:❖\:{\textsf{\textbf{\red{Given:}}}}$

• $\sf{\dfrac{x\:+\:1}{x}}\:=\:2$

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$\large\:❖\:{\textsf{\textbf{\red{Find:}}}}$

• $\sf{\bold{î\succcurlyeq}}\:\:{\dfrac{x²\:+\:1}{x²}}$

• $\sf{\bold{îî\succcurlyeq}}\:\:{\dfrac{x⁴\:+\:1}{x⁴}}$

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$\large\:❖\:{\textsf{\textbf{\red{Solution:}}}}$

• $\sf{\bold{î\succcurlyeq}}\:\:{\dfrac{x²\:+\:1}{x²}}$

Squaring Both Side :

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$\:\:\:\:\:\dashrightarrow\:\sf\bigg (\:{\dfrac{x\:+1}{x}}\:{\bigg )}^{2}\:=\:{(\:2\:)}^{2}$

${ }$

$\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{x²\:+\:1}{x²}}\:+\:2\:=\:4$

• Because (a + b)² = a² + b² + 2ab

${ }$

$\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{x²\:+\:1}{x²}}\:=\:4\:-\:2$

${ }$

$\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{x²\:+\:1}{x²}}\:=\:2$

${ }$

$\:\:\:\:\:\:\therefore\:{\underline{\sf{So,\:the\:Value\:of\:{\dfrac{x²\:+\:1}{x²}}\:is\:{\textsf{\textbf{2}}}}}}.$

${ }$

$\:\:\:\:\:\:\:\:━━━━━━━━━━━━━━━━━━━$

${ }$

• $\sf{\bold{îî\succcurlyeq}}\:\:{\dfrac{x⁴\:+\:1}{x⁴}}$

${ }$

$\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{x²\:+\:1}{x²}}\:=\:2$

Squaring Both Side :

${ }$

$\:\:\:\:\:\dashrightarrow\:\sf\bigg (\:{\dfrac{x²\:+\:1}{x²}}{\:\bigg )}^{2}\:=\:{(\:2\:)}{^ 2}$

• Because (a + b)² = a² + b² + 2ab

${ }$

$\:\:\:\:\:\dashrightarrow\:\sf\:{\dfrac{x⁴\:+\:1}{x⁴}}\:+\:2\:=\:4$

${ }$

$\:\:\:\:\:\dashrightarrow\:\sf\:{\dfrac{x⁴\:+\:1}{x⁴}}\:=\:4\:-\:2$

${ }$

$\:\:\:\:\:\dashrightarrow\:\sf\:{\dfrac{x⁴\:+\:1}{x⁴}}\:=\:2$

${ }$

$\:\:\:\:\:\:\therefore\:{\underline{\sf{So,\:the\:Value\:of\:{\dfrac{x⁴\:+\:1}{x⁴}}\:is\:{\textsf{\textbf{2}}}}}}.$

${ }$

$\:\:\:\:\:\:\:\:━━━━━━━━━━━━━━━━━━━$