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The equation provided is:

\[
4 \cdot 5^{2x} - 29 \cdot 10^x + 25 \cdot 4^x = 0
\]

### Step-by-Step Solution:

1. **Substitute** common bases:
   - Recognize that \(5^{2x} = (5^x)^2\) and \(4^x = (2^2)^x = (2^x)^2\). Thus, we can attempt a substitution.

2. Let \(y = 10^x\). This simplifies the equation using the fact that \(10^x = 2^x \cdot 5^x\), so the equation becomes quadratic.

Would you like me to proceed with solving it this way, or would you like more details on any step?

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**Relative Questions**:
1. How would you approach an equation involving different exponential terms?
2. Can you explain the method of solving a quadratic equation?
3. What are some common strategies for simplifying exponential equations?
4. How does recognizing common bases help in solving exponential equations?
5. What substitution methods are useful in non-linear equations?

**Tip**: Always look for ways to simplify exponential equations by substituting common terms or converting into a quadratic form for easier handling.

Discover how to solve the exponential equation 4 * 5^(2x) - 29 * 10^x + 25 * 4^x = 0 using substitution and quadratic methods. This solution explores algebraic concepts, properties of exponents, and simplifying into a quadratic form. Ideal for high school students in Grades 10-12.

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