Solve the equation using any convenient method.
a2x2 – b2 = 0,   a and b are real numbers
x =
(smaller value when a and b are positive values)
x =
(larger value when a and b are positive values)

The equation given is:

\[
a^2x^2 - b^2 = 0
\]

We can solve this equation using the following steps:

### Step 1: Isolate \(x^2\)
\[
a^2x^2 = b^2
\]

### Step 2: Divide both sides by \(a^2\)
\[
x^2 = \frac{b^2}{a^2}
\]

### Step 3: Take the square root of both sides
\[
x = \pm \frac{b}{a}
\]

This gives two possible solutions for \(x\):

\[
x = \frac{b}{a} \quad \text{or} \quad x = -\frac{b}{a}
\]

### Step 4: Determine the smaller and larger values when \(a > 0\) and \(b > 0\)

- **Smaller value**: \(x = -\frac{b}{a}\)
- **Larger value**: \(x = \frac{b}{a}\)

### Final Answer:

- Smaller value of \(x\): \(-\frac{b}{a}\)
- Larger value of \(x\): \(\frac{b}{a}\)

Would you like any further details on this solution, or do you have any questions?

Here are 5 related questions to consider:
1. How does the solution change if \(a\) or \(b\) are negative?
2. Can this equation be solved graphically? What would the graph look like?
3. What is the significance of the roots in a physical context, such as in motion problems?
4. How does the discriminant confirm the nature of the roots?
5. What happens if \(a = 0\) or \(b = 0\)?

**Tip:** Always check for special cases like zero values or undefined terms when solving equations.

Learn how to solve the equation a^2x^2 - b^2 = 0 with a detailed step-by-step solution. This problem covers algebraic concepts related to quadratic equations and their solutions. Suitable for advanced high school students.

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