Question

if x = √5-2 find the value of (x - 1/x)²​

Answer 1

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Answer 2

Answer:

Given:-

If x = √5 - 2. Find the value of $( x - \dfrac{1}{x} )²$ .

To Find:-

The value of $( x - \dfrac{1}{x} )²$ .

Note:-

●》Here, $( a - \dfrac{1}{a} )² = a² + \dfrac{1}{a²} - 2$

●》( a - b )² = a² + b² - 2ab.

Solution:-

$\huge\red{x = √5 - 2}$

$\huge\red{ \ \ \ \ The \ \ value \ \ of \ \ ( x - \dfrac{1}{x} )² = ?}$

☆ According to note first point~

$( a - \dfrac{1}{a} )² = a² + \dfrac{1}{a²} - 2$

☆ Here, "a" = x~

▪︎$( x - \dfrac{1}{x} )² = x² + \dfrac{1}{x²} - 2$

☆ Applying "x" value ( here we are finding $( x - \dfrac{1}{x} )²$ , so we won't apply "x" value to it )~

▪︎$( x - \dfrac{1}{x} )² = ( √5 - 2 )² + \dfrac{1}{( √5 - 2 )²} - 2$

☆ According to note second point ( here "a" = √5, "b" = 2 )~

▪︎$( x - \dfrac{1}{x} )² = ( √5 )² + ( 2 )² - ( 2 × √5 × 2 ) + \dfrac{1}{(√5 )² + ( 2 )² - ( 2 × √5 × 2 )} - 2$

☆ As Under root will be canceled by square~

▪︎$( x - \dfrac{1}{x} )² = 5 + ( 2 × 2 ) - 4√5 + \dfrac{1}{5 + ( 2 × 2 ) - 4√5} - 2$

▪︎$( x - \dfrac{1}{x} )² = 5 + 4 - 4√5 + \dfrac{1}{5 + 4 - 4√5} - 2$

▪︎$( x - \dfrac{1}{x} )² = 9 - 4√5 + \dfrac{1}{9 - 4√5} - 2$

☆ L.C.M. = 9 - 4√5~

▪︎$( x - \dfrac{1}{x} )² = 9 - 4√5 × \dfrac{9 - 4√5}{9 - 4√5} + \dfrac{1}{9 - 4√5} - 2 × \dfrac{9 - 4√5}{9 - 4√5}$

▪︎$( x - \dfrac{1}{x} )² = \dfrac{( 9 - 4√5 )²}{9 - 4√5} + \dfrac{1}{9 - 4√5} - \dfrac{2 × ( 9 - 4√5 )}{9 - 4√5}$

☆ According to note first point [ for opening ( 9 - 4√5 )² ]

▪︎$( x - \dfrac{1}{x} )² = \dfrac{( 9 )² + ( 4√5 )² - ( 2 × 9 × 4√5 )}{ 9 - 4√5} + \dfrac{1}{9 - 4√5} - \dfrac{2 × 9 - 2 × 4√5}{9 - 4√5}$

▪︎$( x - \dfrac{1}{x} )² = \dfrac{( 9 × 9 ) + ( 4√5 )² - 72√5}{9 - 4√5} + \dfrac{1}{9 - 4√5} \dfrac{- 18 - 8√5}{9 - 4√5}$

☆ After multiplying sign in bracket~

▪︎$( x - \dfrac{1}{x} )² = \dfrac{81 + 16 × 5 - 72√5}{9 - 4√5} + \dfrac{1}{9 - 4√5} \dfrac{- 18 + 8√5}{9 - 4√5}$

▪︎$( x - \dfrac{1}{x} )² = \dfrac{81 + 80 - 72√5}{9 - 4√5} + \dfrac{1}{9 - 4√5} \dfrac{- 18 + 8√5}{9 - 4√5}$

▪︎$( x - \dfrac{1}{x} )² = \dfrac{161 - 72√5}{9 - 4√5} + \dfrac{1}{9 - 4√5} \dfrac{- 18 + 8√5}{9 - 4√5}$

▪︎$( x - \dfrac{1}{x} )² = \dfrac{162 - 72√5}{9 - 4√5} \dfrac{- 18 + 8√5}{9 - 4√5}$

☆ After subtracting common terms~

▪︎$( x - \dfrac{1}{x} )² = \dfrac{144 - 64√5}{9 - 4√5}$

▪︎$( x - \dfrac{1}{x} )² = \dfrac{16 (9 - 4√5 )}{9 - 4√5}$

☆ After dividing~

▪︎$( x - \dfrac{1}{x} )² = 16$

$\huge\pink{The \ \ value \ \ of \ \ ( x - \dfrac{1}{x} )² = 16}$

Answer:-

Hence, The value of $( x - \dfrac{1}{x} )² = 16$ .

:)