To solve the equation \(x^2 - 529 = 0\) by factoring, we start by rewriting it in a form that can be factored:

### Step 1: Rewrite the Equation

Given equation:

\[
x^2 - 529 = 0
\]

### Step 2: Recognize as a Difference of Squares

The expression \(x^2 - 529\) is a difference of squares because it can be written as:

\[
x^2 - 529 = (x - \sqrt{529})(x + \sqrt{529})
\]

Since \(\sqrt{529} = 23\), we can rewrite this as:

\[
x^2 - 529 = (x - 23)(x + 23)
\]

### Step 3: Set Each Factor to Zero

Now, set each factor equal to zero:

\[
x - 23 = 0 \quad \text{or} \quad x + 23 = 0
\]

### Step 4: Solve for \(x\)

Solving each equation:

\[
x = 23 \quad \text{or} \quad x = -23
\]

### Solution

The solutions to the equation are:

\[
x = 23 \quad \text{and} \quad x = -23
\]

Would you like a detailed explanation of any step, or do you have any questions?

### Related Questions

1. How do you factor other types of quadratic equations?
2. What is the difference of squares, and how is it used in factoring?
3. Can all quadratic equations be factored easily?
4. How do you solve quadratic equations by completing the square?
5. How does factoring compare to using the quadratic formula?

### Tip

Always look for patterns like the difference of squares when factoring equations; it simplifies the process significantly!

Learn how to solve the equation x² - 529 = 0 by factoring. This problem involves recognizing the difference of squares and applying factoring techniques to find the solutions. Suitable for high school students studying algebra and quadratic equations.

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