x^2 + 1/x^2 = 66.  Then find the value of x-1/x+2=?

We are given the equation:

\[
x^2 + \frac{1}{x^2} = 66
\]

To solve for \( x - \frac{1}{x} + 2 \), we will follow these steps:

### Step 1: Express \( x^2 + \frac{1}{x^2} \) in terms of \( \left( x - \frac{1}{x} \right)^2 \)
We know that:

\[
\left( x - \frac{1}{x} \right)^2 = x^2 - 2 + \frac{1}{x^2}
\]

This can be rearranged as:

\[
x^2 + \frac{1}{x^2} = \left( x - \frac{1}{x} \right)^2 + 2
\]

We are given \( x^2 + \frac{1}{x^2} = 66 \), so:

\[
\left( x - \frac{1}{x} \right)^2 + 2 = 66
\]

Subtract 2 from both sides:

\[
\left( x - \frac{1}{x} \right)^2 = 64
\]

### Step 2: Solve for \( x - \frac{1}{x} \)
Taking the square root of both sides:

\[
x - \frac{1}{x} = \pm 8
\]

### Step 3: Find \( x - \frac{1}{x} + 2 \)
For both possible values of \( x - \frac{1}{x} \):

1. If \( x - \frac{1}{x} = 8 \), then:
   \[
   x - \frac{1}{x} + 2 = 8 + 2 = 10
   \]

2. If \( x - \frac{1}{x} = -8 \), then:
   \[
   x - \frac{1}{x} + 2 = -8 + 2 = -6
   \]

### Final Answer:
The possible values for \( x - \frac{1}{x} + 2 \) are **10** and **-6**.

Would you like further details or have any questions?

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Here are 5 related questions to explore further:

1. How would the solution change if the original equation were \( x^2 + \frac{1}{x^2} = 18 \)?
2. What if the equation were \( x^2 - \frac{1}{x^2} = 40 \)? How would you solve it?
3. Can you derive a general formula for \( x + \frac{1}{x} \) given \( x^2 + \frac{1}{x^2} \)?
4. How would you solve for \( x + \frac{1}{x} \) if \( x - \frac{1}{x} = 5 \)?
5. What if you were given \( x^2 + \frac{1}{x^2} = 0 \)? What conclusions can you draw?

**Tip:** When solving equations involving symmetrical expressions like \( x \) and \( \frac{1}{x} \), try expressing them in terms of a simpler variable like \( y = x - \frac{1}{x} \) or \( z = x + \frac{1}{x} \). This often simplifies the problem significantly.

Learn how to solve the equation x^2 + 1/x^2 = 66 and find the value of x - 1/x + 2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic equations, algebraic manipulation, and square roots. Suitable for students in Grades 10-12.

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