Question

$if \: x = \frac{ \sqrt{5} - 2 }{ \sqrt{5} + 2} find \: the \: value \: of \: x + \frac{1}{x}$
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Answer 1

18

Step-by-step explanation:

Given:

$x = \dfrac{ \sqrt{5} - 2 }{ \sqrt{5} + 2}$

To Find:

$x + \dfrac{1}{x}$

Process to Solve:

1. Don't Need to Rationalize because it's a like fraction

2. Take L.C.M

3. Use Algebra Formula

4. Simplify and Get the answer

Simple, Isn't it ?

Solution:

$x + \frac{1}{x} \\ \\ = \frac{ \sqrt{5} - 2}{ \sqrt{5} + 2} + \frac{ \sqrt{5} + 2}{ \sqrt{5} - 2} \\ \\ [\text{By taking L.C.M}] \\ \\ = \frac{ ( \sqrt{5} - 2) {}^{2} + {( \sqrt{5} + 2) }^{2} }{( \sqrt{5} + 2)( \sqrt{5} - 2) } \\ \\ = \frac{2 \{(\sqrt{5 }) {}^{2} + (2 {)}^{2} \} }{ {( \sqrt{5})}^{2} - {(2)}^{2} } \\ \\ = \frac{2(5 + 4)}{5 - 4} \\ \\ = \frac{2(9)}{1} \\ \\ = 18$

Therefore, The required answer is 18.

Algebra Formula Used

a² - b² = (a + b)(a - b)

(a - b)² + (a + b)² = 2(a² + b²)

What is Like Fraction ?

When the nominator and denominator of two fraction are similar but different by sign.