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.