Question

X+1/X=√5and find the value of x^2+1/x^2
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Answer 1

Answer:

3

Step-by-step explanation:

As per the provided information in the given question, we have:

⠀⠀⠀⠀⠀★ $\rm { x + \dfrac{1}{x} = \sqrt{5}}$

We've been asked to calculate the value of :

⠀⠀⠀⠀⠀★ $\bf { x^2 + \dfrac{1}{x^2}}$

According to the given question the sum of x and its reciprocal is √5. That is,

$\twoheadrightarrow \rm{\quad {x + \dfrac{1}{x} = \sqrt{5} }}$

Squaring both sides. We get,

$\twoheadrightarrow \rm{\quad {{ \Bigg ( x + \dfrac{1}{x}\Bigg )}^{2} =( \sqrt{5})^2 }}$

Writing the square to the term in RHS. Using the identity in the LHS :

⠀⠀⠀⠀⠀⠀★ (a + b)² = a² + b² + 2ab

$\twoheadrightarrow \rm{\quad { x^2 + {\Bigg (\dfrac{1}{x}\Bigg )}^{2} + 2\Bigg ( \not{x} \times \dfrac{1}{\not{x}} \Bigg ) = 5 }}$

As, we know that (a/b)² = a²/b². So, missing this identity rearranging the terms in LHS and performing multiplication in the brackets.

$\twoheadrightarrow \rm{\quad { x^2 + \dfrac{1}{x^2} + 2\Big ( 1 \Big ) = 5 }}$

Performing multiplication in LHS.

$\twoheadrightarrow \rm{\quad { x^2 + \dfrac{1}{x^2} + 2 = 5 }}$

Transposing 2 from LHS to RHS. Its sign will get changed.

$\twoheadrightarrow \rm{\quad { x^2 + \dfrac{1}{x^2} = 5 - 2 }}$

Performing subtraction in RHS to find the required answer.

$\twoheadrightarrow \quad\underline{\boxed { \bf {x^2 + \dfrac{1}{x^2} = 3 }}}$

❝ Therefore, the required answer is 3.❞