https://file.aiquickdraw.com/mathbot/file/172680128693154pnn215.jpg

The equation given in the image is:

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

This looks like a quadratic equation in terms of \( 5^x \). To solve it, we can apply a substitution method:

### Step 1: Substitution
Let \( y = 5^x \). Then:
\[
5^{2x} = (5^x)^2 = y^2
\]
The equation now becomes:
\[
y^2 - 10y + 25 = 0
\]

### Step 2: Solve the quadratic equation
We now solve the quadratic equation:
\[
y^2 - 10y + 25 = 0
\]
This is a perfect square trinomial and can be factored as:
\[
(y - 5)^2 = 0
\]

Thus, the solution for \( y \) is:
\[
y = 5
\]

### Step 3: Back-substitute \( y = 5^x \)
Now, we return to the original variable \( x \). Since \( y = 5^x \) and we found that \( y = 5 \), we have:
\[
5^x = 5
\]

This implies:
\[
x = 1
\]

### Final Answer:
\[
x = 1
\]

Would you like further clarification or details?

Here are 5 related questions:
1. How can you solve equations involving logarithms and exponents?
2. What happens if the quadratic equation cannot be factored easily?
3. Can substitution always simplify exponential equations?
4. How does this approach work for exponential equations with bases other than 5?
5. What are some real-life applications of exponential equations?

**Tip**: When solving equations with exponents, substitution can often simplify the problem into a quadratic form, making it easier to solve.

Solve the equation for x: 5^(2x) - 10 * 5^x + 25 = 0

Learn how to solve the exponential equation 5^(2x) - 10 * 5^x + 25 = 0 by using the substitution method. This problem involves key mathematical concepts such as exponential equations, quadratic equations, and properties of exponents. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation