Solve: If x = 5 – 2√6, find 1/x, (x2 – 1/x2):

Solve: If x = 5 – 2√6, find 1/x, (x² – 1/x²)

Solution:

Given:

x = 5 – 2√6

so,

1/x = 1/(5 – 2√6)

By rationalizing the denominator,

1/(5 – 2√6) = [1(5 + 2√6)] / [(5 – 2√6) (5 + 2√6)]

= [(5 + 2√6)] / [5² – (2√6)²]

= [(5 + 2√6)] / [25 – 4.6]

= [(5 + 2√6)] / [25 – 24]

= (5 + 2√6)

Then,

x + 1/x = 5 – 2√6 + (5 + 2√6)

= 10

Square on both sides we get

(x + 1/x)² = 10²

x² + 1/x² + 2x.1/x = 100

x² + 1/x² + 2 = 100

x² + 1/x² = 100 – 2

= 98

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