Question

If x= 2 + √5 , find the value of x2-1/x2
)
(A) 2√5
(B) 4√5
(C) 6√5
(D) 8√5​

Answer 1

Solution :-

Given ,

x = 2 + √5

We need to find ,

x² - (1/x²)

Firstly finding x² & 1/x²

x = 2 +√5

Squaring on both sides

→ x² = (2 + √5)²

→ x² = 2² + (√5)² + 2(2)(√5)

→ x² = 4 + 5 + 4√5

→ x² = 9 + 4√5

Now , finding 1/x²

→ 1/x² = 1/9 + 4√5

______________________

Now , finding the question

→ (9 + 4√5) - [1/(9 + 4√5)]

→ [(9 + 4√5)² - 1/(9 + 4√5)]

→ 9² + 16(5) - 1/ 9 + 4√5

→ 18 + 80 - 1/9 + 4√5

→ 97/9 + 4√5

Hence , x² - 1/x² = 97/9 + 4√5.

Answer 2

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$\huge\sf\red{Given \::-}$

• $\large\sf\orange{x\:= \:2 +\sqrt{5}}$

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$\huge\sf\red{To\:Find \::-}$

• $\large\sf\orange{x^{2} -\bigg(\dfrac{1}{x^{2}}\bigg)}$

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$\huge\sf\red{Solution\::-}$

$\large\sf\underline\blue{At\: first,\:We \:find\: x^{2} \:\&\:\dfrac{1}{x^2}}$

$→\:\sf\purple{x\:= \:2\: +\sqrt{5}}$

$\sf\gray{(Squaring\: on\:both\:sides)}$

$→\:\sf\green{x^2 \:= \:(2 \:+\sqrt{5})^2 }$

$→\:\sf\purple{x^2 \:=\: 2^{2} \:+\:(\sqrt{5})^{2}\:2(2)(\sqrt{5})}$

$→\:\sf\green{x^2 \:= \:4\:+\:5\:+\:4\sqrt{5}}$

$→\:{\boxed{\sf{\purple{x^{2}\:=\:9\:+\:4\sqrt{5}}}}}$

$\sf\underline\orange{Putting\:x^{2}\:value\:in\:\frac{1}{x^2}}$

$\mapsto\:\sf\purple{\dfrac{1}{x^2}}$

$\mapsto\:\sf\green{\dfrac{1}{9\:+\:4\sqrt{5}^2} }$

$\sf\underline\orange{Now, Finding\:the\:value\: x^{2} -\bigg(\dfrac{1}{x^{2}}\bigg) }$

$\hookrightarrow\:\sf\blue{(9\:+\:4\sqrt{5})\:-\:\bigg( \dfrac{1}{9\:+\:4\sqrt{5}} \bigg)}$

$\hookrightarrow\:\sf\gray{9^{2}\:+\:16(5) \:-\: \dfrac{1}{9\:+\:4\sqrt{5}} }$

$\hookrightarrow\:\sf\blue{18\:+\:80\:-\:\dfrac{1}{9\:+\:4\sqrt{5}} }$

$\hookrightarrow\:{\boxed{\sf{\pink{\dfrac{97}{9\:+\:4\sqrt{5}}}}} }$

$\Large\sf\orange{Hence,}$

$\star\:\sf\red{The \ value \ of \ x^{2} -\bigg(\dfrac{1}{x^{2}}\bigg) \: is \: \dfrac{97}{9\:+\:4\sqrt{5}}}$

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