To simplify and solve the equation \((x + a)(x - a) + a^2 - x^2 = 0\), let's break it down step by step:

1. **Expand** the first term using the difference of squares identity:  
   \[
   (x + a)(x - a) = x^2 - a^2
   \]

2. **Substitute** this into the equation:
   \[
   x^2 - a^2 + a^2 - x^2 = 0
   \]

3. **Combine like terms**:
   \[
   x^2 - x^2 - a^2 + a^2 = 0
   \]

   The \(x^2\) terms and the \(a^2\) terms cancel each other out, leaving:
   \[
   0 = 0
   \]

Thus, the equation simplifies to \(0 = 0\), which is **always true** for any value of \(x\) and \(a\). This implies that the equation is an identity.

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Would you like more details on any step or have any questions?

Here are five related questions to expand on this:

1. What are other examples of identities similar to this one?
2. How does the difference of squares formula work in general?
3. Can this technique be used to factor more complex polynomials?
4. What happens if we replace \(a^2\) with a different constant in the equation?
5. How would the result change if \(x^2\) had a different coefficient?

**Tip:** Recognizing special patterns like the difference of squares can simplify many algebraic expressions.

Discover how to solve the identity equation (x + a)(x - a) + a^2 - x^2 = 0 using the difference of squares formula. This algebraic solution demonstrates key concepts like identity equations and simplification. Perfect for high school students in Grades 9-12.

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