Question

please solve this question and give a solution

Answer 1

x = √(8 + 4√3)

x² = (8 + 4√3) -------(1)

again,

1/x = 1/√(8 +4√3)

= √(8 - 4√3)/√(8 +4√3).√(8 -4√3)

= √(8 - 4√3)/√(8² -4√3² )

=√(8 -4√3)/√(64-48)

= √(8 - 4√3)/4

1/x = √(8 -4√3)/4

4/x = √(8 - 4√3)

16/x² = (8 - 4√3) -------(2)

now,

(x - 4/x ) = √(x² + 16/x² -2.x.4/x )

= √( 8 +4√3 + 8 -4√3 -8)

= √8 = 2√2

hence ,
x - 4/x = 2√2

Answer 2

x = √(8 + 4√3)

x² = (8 + 4√3)

Rationalise the denominator,

Rationalising factor = √8-4√3

1/x = 1/√(8 +4√3)

1/x = √(8 - 4√3)/√(8 +4√3)×√(8 -4√3)

1/x = √(8 - 4√3)/√(8² -{4√3}²)

1/x = √(8-4√3)/(64-16(3))

1/x =√(8 - 4√3)/√(64-48)

1/x = √(8-4√3)/√16

1/x = √(8 - 4√3)/4

1/x = √(8 -4√3)/4

4/x = √(8 - 4√3)

16/x² = (8 - 4√3)

We need to find (x-4/x):-

(x - 4/x ) = √(x² + 16/x² -2(x)(4/x))

= √( 8+4√3+8-4√3 -8)

= √16-8 =√8 = 2√2

Therefore, x - 4/x = 2√2

Option (B) is correct.

Hope it helps