Question

If x = √5 +2, then x - 1/x equals

Answer 1

Question:

$If \: x = \sqrt{5} +2, then \: x - \frac{1}{x} \: equals.$

To find:

$To \: find \: the \: value \: of \\ \big(x - \frac{1}{x}\big)$

Given:

$x = \sqrt{5}+2$

Answer:

$The \: value \: is \:\boxed{\sf\pink{ \frac{ \sqrt{5} - 3 }{2} }}$

Step-by-step explanation:

$x = \sqrt{5}+2$

$\implies x - \frac{1}{x}$

$\implies \sqrt{5} + 2 - \frac{1}{ \sqrt{5} + 2 }$

$= \frac{ {( \sqrt{5} + 2) }^{2} - 1}{ \sqrt{5} + 2 }$

$= \frac{5 + 4 \sqrt{5} + 4 - 1 }{ \sqrt{5} + 2}$

$= \frac{ 4\sqrt{5} +3 + 5 }{ \sqrt{5} + 2 }$

$= \frac{ 4\sqrt{5} +8 }{ \sqrt{5} + 2 }$

$= \frac{2(2 \sqrt{5} + 4) }{2( \sqrt{5} + 1 }$

$= \frac{2 \sqrt{5} + 4 }{ \sqrt{5} + 1 }$

Rationalising by √5 - 1

$= \frac{(2 \sqrt{5} + 4)( \sqrt{5} - 1) }{( \sqrt{5} + 1)( \sqrt{5} - 1) }$

$= \frac{10 - 2 \sqrt{5} + 4 \sqrt{5} - 4 }{ {( \sqrt{5}) }^{2} - {1}^{2} }$

$= \frac{2 \sqrt{5} - 6 }{5 - 1}$

$= \frac{2 \sqrt{5} - 6 }{4}$

$= \frac{2( \sqrt{5} - 6) }{2(2)}$

$\implies \boxed{\sf\pink{ \frac{ \sqrt{5} - 3 }{2} }}$

$\implies \boxed{ x - \frac{1}{x} = \frac{ \sqrt{5} - 3 }{2} }$

Answer 2

Answer:

√5+2 - 1/√5+2×√5-2/√5-2

(a+b) (a-b)= (a)²-(B)²

√5+2-√5+2/(√5)²-(2)²

√5+2-√3-2/5-4

√5+2-√5-2

2√5 is Ans.... thanku