If x = √6 + 2 and y = √6 - 2, then what is the value of (x/y + y/x)^2 - 3?

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Given x = √6 + 2 and y = √6 - 2.

We need to find the value of (x/y + y/x)^2 - 3.

Step 1: Simplify the expression inside the square.

(x/y + y/x) = (x^2 + y^2) / (xy)

Step 2: Calculate x^2, y^2, and xy.

x^2 = (√6 + 2)^2 = (√6)^2 + 2*2*√6 + 2^2 = 6 + 4√6 + 4 = 10 + 4√6

y^2 = (√6 - 2)^2 = (√6)^2 - 2*2*√6 + 2^2 = 6 - 4√6 + 4 = 10 - 4√6

xy = (√6 + 2)(√6 - 2) = (√6)^2 - 2^2 = 6 - 4 = 2

Step 3: Calculate numerator x^2 + y^2.

x^2 + y^2 = (10 + 4√6) + (10 - 4√6) = 10 + 10 + 4√6 - 4√6 = 20

Step 4: Calculate (x/y + y/x).

(x/y + y/x) = (x^2 + y^2) / (xy) = 20 / 2 = 10

Step 5: Calculate (x/y + y/x)^2 - 3.

= 10^2 - 3 = 100 - 3 = 97

Therefore, the value is 97.

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