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The given mathematical problem can be translated as follows:

We are provided the equation \( y = a^{2x} + a^{-2x} - 2(a^x + a^{-x}) + 2 \), where \( a > 0 \) and \( a \neq 1 \), and we define \( a^x + a^{-x} = t \). 

We are tasked with expressing \( y \) in terms of \( t \) and finding the minimum value of \( y \).

### Step 1: Express \( y \) in terms of \( t \)
Since we are given \( t = a^x + a^{-x} \), let’s try to manipulate \( y \) using this.

\[
y = a^{2x} + a^{-2x} - 2(a^x + a^{-x}) + 2
\]

We know that:

\[
t = a^x + a^{-x}
\]

Now we also need to express \( a^{2x} + a^{-2x} \) in terms of \( t \). Recall the identity:

\[
a^{2x} + a^{-2x} = (a^x + a^{-x})^2 - 2
\]

Using this identity, we can substitute:

\[
y = (a^x + a^{-x})^2 - 2 - 2(a^x + a^{-x}) + 2
\]

### Step 2: Simplify the expression
Substitute \( t \) into the equation:

\[
y = t^2 - 2 - 2t + 2
\]

Simplify the constants:

\[
y = t^2 - 2t
\]

### Step 3: Minimize \( y = t^2 - 2t \)
To find the minimum value of \( y \), we can treat this as a simple quadratic function:

\[
y = t^2 - 2t
\]

This is a standard quadratic expression. The minimum value of a quadratic \( y = ax^2 + bx + c \) occurs at \( t = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -2 \), so the minimum occurs at:

\[
t = \frac{-(-2)}{2(1)} = 1
\]

### Step 4: Compute the minimum value of \( y \)
Substitute \( t = 1 \) into the equation for \( y \):

\[
y = (1)^2 - 2(1) = 1 - 2 = -1
\]

Thus, the minimum value of \( y \) is \( \boxed{-1} \).

Would you like more details or clarifications?

Here are 5 related questions to explore:
1. How do you derive identities like \( a^{2x} + a^{-2x} = (a^x + a^{-x})^2 - 2 \)?
2. What is the geometric interpretation of minimizing quadratic functions?
3. Can you apply this approach to other types of exponential functions?
4. How does the constraint \( a > 0 \) and \( a \neq 1 \) affect the behavior of the function?
5. What happens to the function \( y \) if we vary \( a \)?

**Tip:** When simplifying complex exponential expressions, look for opportunities to use common identities or substitutions like \( t = a^x + a^{-x} \). This can greatly simplify your work!

a > 0, a ≠ 1 とする。関数 y = a^(2x) + a^(-2x) − 2(a^x + a^(-x)) + 2 について、a^x + a^(-x) = t とおく。y を t を用いて表し, y の最小値を求めよ。

Solve the problem where y = a^(2x) + a^(-2x) − 2(a^x + a^(-x)) + 2, using the substitution t = a^x + a^(-x). This involves simplifying the expression and finding the minimum value of y, expressed in terms of t. Perfect for high school students exploring exponential and quadratic functions.

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