Solution:

To find the value of x^2 + 1/x^2, we need to first find the value of x.

Given: x + 1/x = 11

Step 1: Rearrange the given equation

To simplify the equation, we can multiply both sides by x:

x(x + 1/x) = 11x

Simplifying further, we get:

x^2 + 1 = 11x

Step 2: Rearrange the equation as a quadratic equation

Now, let's rearrange the equation to get a quadratic equation:

x^2 - 11x + 1 = 0

Step 3: Solve the quadratic equation

To solve the quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

For our quadratic equation, the values of a, b, and c are:

a = 1
b = -11
c = 1

Plugging in these values into the quadratic formula, we get:

x = (-(-11) ± √((-11)^2 - 4(1)(1))) / (2(1))

Simplifying further, we have:

x = (11 ± √(121 - 4)) / 2

x = (11 ± √117) / 2

x = (11 ± √(9 × 13)) / 2

x = (11 ± 3√13) / 2

Therefore, the two possible values of x are:

x1 = (11 + 3√13) / 2
x2 = (11 - 3√13) / 2

Step 4: Find the value of x^2 + 1/x^2

Now that we have the values of x, we can substitute them into the expression x^2 + 1/x^2 to find the final answer.

For x1 = (11 + 3√13) / 2:

x1^2 + 1/x1^2 = [(11 + 3√13) / 2]^2 + 1/[(11 + 3√13) / 2]^2

Simplifying this expression will give us the value of x1^2 + 1/x1^2.

Similarly, for x2 = (11 - 3√13) / 2:

x2^2 + 1/x2^2 = [(11 - 3√13) / 2]^2 + 1/[(11 - 3√13) / 2]^2

Simplifying this expression will give us the value of x2^2 + 1/x2^2.

Hence, the value of x^2 + 1/x^2 can be found by substituting the values of x1 and x2 into the respective expressions and simplifying the equations obtained.