Question

If x=4-√5find[x-1/x]^2

Answer 1

Hi.

$\text{Here, Given x = 4 - \sqrt{5} }\\\\\text{Squaring Both Sides,}\\\\x^2 = (4 - \sqrt{5})^2 = 4^2 + (\sqrt{5})^2 - 2(4)(\sqrt{5})\\\\\implies x^2 = 16 + 5 - 8 \sqrt{5} = 21 - 8 \sqrt{5}\\\\\text{Now,}\\\\(\dfrac{1}{x})^2 = \dfrac{1^2}{x^2} = \dfrac{1}{x^2} = \dfrac{1}{21 - 8 \sqrt{5}}\\\\\\= \dfrac{1}{21 - 8 \sqrt{5}} \times \dfrac{21 + 8 \sqrt{5}}{21 + 8 \sqrt{5}}\\\\\\= \dfrac{21 + 8 \sqrt{5}}{(21)^2 - (8 \sqrt{5})^2}\\\\\\= \dfrac{21 + 8 \sqrt{5}}{441 - 320}$

$\dfrac{21 + 8 \sqrt{5}}{121}\\\\\\\text{Now,}\\\\(x - \dfrac{1}{x})^2 = (x)^2 + (\dfrac{1}{x})^2 - 2(x)(\dfrac{1}{x})\\\\= (21 - 8 \sqrt{5}) + (\dfrac{21 + 8 \sqrt{5}}{121}) - 2\\\\\\= \dfrac{2541 + 21 - 968 \sqrt{5} + 8 \sqrt{5} - 242}{121}\\\\\\= \large \underline{\boxed{\boxed{\dfrac{2320 - 960 \sqrt{5}}{121}}}}$

Hope it helps.

Answer 2

Hello Mate!

x = 4 - √5

1 / x = 1 / ( 4 - √5 )

= 1 / ( 4 - √5 ) × ( 4 + √5 )/( 4 + √5 )

= ( 4 + √5 )/( 4² - √5² )

= ( 4 + √5 )/( 16 - 5 ) = ( 4 + √5 )/11

So, ( x - 1 / x )²

= [ 4 - √5 - ( 4 + √5 )/11 ]²

= [ ( 44 - 11√5 )/11 - ( 4 + √5 )/11 ]²

= [ ( 44 - 11√5 - 4 - √5 )/11]²

= [ ( 40 - 12√5 )/11 ]²

= ( 1600 + 144 × 5 - 960√5 )/121

= ( 1600 + 720 - 960√5 )/121

= ( 2320 - 960√5 )/121

Have great future ahead!