To solve the equation (x + 1/x) = 6, we need to find the value of x^2 + 1/x^2. Let's solve it step by step.

Step 1: Multiply the entire equation by x to eliminate the fraction:
x(x + 1/x) = 6x
This simplifies to:
x^2 + 1 = 6x

Step 2: Move all terms to one side of the equation to get a quadratic equation:
x^2 - 6x + 1 = 0

Step 3: Solve the quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = -6, and c = 1.

Plugging in the values, we get:
x = (-(-6) ± √((-6)^2 - 4(1)(1))) / (2(1))
x = (6 ± √(36 - 4)) / 2
x = (6 ± √32) / 2
x = (6 ± 4√2) / 2

Step 4: Simplify the expression:
x = 3 ± 2√2

Now, we need to find the value of x^2 + 1/x^2:
x^2 + 1/x^2 = (3 ± 2√2)^2 + 1/(3 ± 2√2)^2

Expanding the square:
x^2 + 1/x^2 = 9 ± 12√2 + 8 + 1/(9 ± 12√2)

Simplifying further, we get:
x^2 + 1/x^2 = 17 ± 12√2 + 1/(9 ± 12√2)

Now, let's consider the positive root, x = 3 + 2√2:
x^2 + 1/x^2 = 17 + 12√2 + 1/(9 + 12√2)

Rationalizing the denominator:
x^2 + 1/x^2 = 17 + 12√2 + (1 * (9 - 12√2)) / ((9 + 12√2) * (9 - 12√2))
x^2 + 1/x^2 = 17 + 12√2 + (9 - 12√2) / (81 - (12√2)^2)
x^2 + 1/x^2 = 17 + 12√2 + (9 - 12√2) / (81 - 144)
x^2 + 1/x^2 = 17 + 12√2 + (9 - 12√2) / -63
x^2 + 1/x^2 = 17 + 12√2 - (12√2 - 9) / 63
x^2 + 1/x^2 = 17 + 12√2 - 12√2 + 9 / 63
x^2 + 1/x^2 = 26 / 63

Simplifying further:
x^