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To explain the presence of \(-2\) in the solution, let's go through the steps carefully:

The problem begins with the equation:

\[
4^x + 4^{-x} = 7
\]

### Step 1: Expressing the equation in terms of \((2^x + 2^{-x})\)
We know that:

\[
4^x = (2^x)^2 \quad \text{and} \quad 4^{-x} = (2^{-x})^2
\]

So, the equation can be rewritten as:

\[
(2^x + 2^{-x})^2 - 2 = 7
\]

### Step 2: Understanding the \(-2\)
The reason for the \(-2\) is based on the algebraic identity for squaring a binomial. When we square \((2^x + 2^{-x})\), it expands as:

\[
(2^x + 2^{-x})^2 = 2^x^2 + 2 \cdot 2^x \cdot 2^{-x} + 2^{-x}^2 = 4^x + 2 + 4^{-x}
\]

So, in the expression for \(4^x + 4^{-x}\), we must subtract \(2\) to match the equation \(4^x + 4^{-x}\).

Thus, the equation becomes:

\[
(2^x + 2^{-x})^2 - 2 = 7
\]

This is where the \(-2\) comes from—it's the correction term to account for the middle term when squaring.

Would you like further clarification on this or other steps? Here are some related questions to explore:

1. How is \((2^x + 2^{-x})^2\) expanded in full?
2. Why do we subtract 2 when transforming the equation?
3. How can we solve for \(2^x + 2^{-x}\) after simplifying the equation?
4. How is the expression \(8^x + 8^{-x}\) linked to \(2^x + 2^{-x}\)?
5. What are some common algebraic identities used in these transformations?

**Tip**: Always remember the binomial expansion formula \((a+b)^2 = a^2 + 2ab + b^2\) when solving similar problems.

Explore the detailed solution for finding the value of -2 in the equation 4^x + 4^{-x} = 7. Learn how algebraic identities and binomial expansion are used to simplify and solve exponential equations. Ideal for students in Grades 10-12.

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