Question

lf x= 5-2 √6find of x+1/x (ii)x-1/x​

Answer 1

$\mathfrak{\large{\underline{\underline{Answer :-}}}}$

x + 1/x = 10; x - 1/x = -4√6

$\mathfrak{\large{\underline{\underline{Explanation :-}}}}$

Given :-

$x = 5 - 2 \sqrt{6}$

To find : x + 1/x ; x - 1/x

Solution :-

First find the value of 1/x

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

To find the value of 1/x we need to rationalise it.

The Rationalising factor of 5 - 2√6 is 5 + 2√6. So multiply numerator and denominator with rationalising factor.

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

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

$\boxed{\sf{(a+b)(a-b)={a}^{2}-{b}^{2}}}$

$= \frac{5 + 2 \sqrt{6} }{25 - 4(6)}$

$= \frac{5 + 2 \sqrt{6} }{25 - 24}$

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

$\boxed{\sf{\bold{\frac{1}{x} = 5 + 2 \sqrt{6} }}}$

Now you can can find the values of x + 1/x ; x - 1/x

1)

$x + \frac{1}{x}$

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

$= 5 - 2 \sqrt{6} + 5 + 2 \sqrt{6}$

$= 10$

$\boxed{\sf{\bold{x+ \frac{1}{x} = 10 }}}$

2)

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

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

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

$= - 4 \sqrt{6}$

$\boxed{\sf{\bold{x- \frac{1}{x} = -4 \sqrt{6} }}}$

Identity used in the problem:-

(a + b) = a² - b²

Extra info related to the question.

What is rationalising factor?

If the product of two irrational numbers is a rational number then each of the the two is rationalising factor of the other.

What is an identity?

An equation is called an identity if it is satisfied by any value that replaces its variables.

Some Important Identities:-

1] (a + b)² = a² + 2ab + b²

2] (a - b)² = a² - 2ab + b²

3] (a + b)(a - b) = a² - b²

4] (x + a)(x + b) = x² + (a + b)x + ab

What is an equation?

An open sentence which contains "=" sign is called an equation.

Example: x + y = 2

2x² + x = 10

What is an open sentence?

A sentence for which we cannot decide whether it is true or false.