The given equation is:

\[
4^x + 4^x = 1
\]

### Step-by-Step Solution:

1. First, observe that the left-hand side consists of two identical terms \(4^x\). So, we can rewrite the equation as:
   \[
   2 \cdot 4^x = 1
   \]

2. Next, divide both sides of the equation by 2:
   \[
   4^x = \frac{1}{2}
   \]

3. Now, express \(4^x\) as \((2^2)^x\), which simplifies to \(2^{2x}\):
   \[
   2^{2x} = \frac{1}{2}
   \]

4. Rewrite \(\frac{1}{2}\) as \(2^{-1}\):
   \[
   2^{2x} = 2^{-1}
   \]

5. Since the bases are the same (both are powers of 2), equate the exponents:
   \[
   2x = -1
   \]

6. Solve for \(x\):
   \[
   x = -\frac{1}{2}
   \]

### Final Answer:
\[
x = -\frac{1}{2}
\]

Would you like further clarification or additional details?

Here are 5 related questions to expand on this:

1. How do you solve exponential equations with different bases?
2. What are the properties of exponents and how do they apply in equations?
3. How would the equation change if you had \(4^x + 2^x = 1\)?
4. What are some common strategies for solving equations involving exponents?
5. Can logarithms be used to solve equations like \(4^x + 4^x = 1\)?

**Tip:** When dealing with exponential equations, always look for ways to simplify terms with the same base before solving.

Discover how to solve the exponential equation 4^x + 4^x = 1 with a step-by-step guide. This problem involves key concepts like exponentiation, properties of exponents, and solving equations by matching powers. Suitable for students in Grades 9-10.

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