Thus, the factored form is \(\boxed{(x - 3)(x + 3)}\).

["Understanding the Factored Form: Why It Is (\boxed{(x - 3)(x + 3)})", "When solving quadratic equations, recognizing and using factored form is a powerful technique that simplifies equations and reveals key insights—like roots and zeros. One of the clearest examples is the expression (\boxed{(x - 3)(x + 3)}), which represents a factored quadratic equation. But what exactly does this factored form mean, and why is it so important? Let’s explore.", "### What Is Factored Form?", "Factored form expresses a quadratic equation as a product of two binomials. In the case of (\boxed{(x - 3)(x + 3)}), this form shows the product of ((x - 3)) and ((x + 3)). When multiplied out, this gives:", "[\n(x - 3)(x + 3) = x^2 - 9\n]", "This matches the standard quadratic form (ax^2 + bx + c) with (a = 1), (b = 0), and (c = -9). The factored form is often simpler to work with in factoring, solving, and factoring cubic or higher-degree polynomials.", "### Why ((x - 3)(x + 3)) Represents a Special Case", "The expression (\boxed{(x - 3)(x + 3)}) is a classic example of the difference of squares, a key algebraic identity. This identity states that:", "[\n(a - b)(a + b) = a^2 - b^2\n]", "Substituting (a = x) and (b = 3), we get:", "[\n(x - 3)(x + 3) = x^2 - 9\n]", "This confirms that ((x - 3)(x + 3)) expands exactly to (x^2 - 9). More importantly, it reveals the roots of the quadratic equation: setting ((x - 3)(x + 3) = 0), we find (x = 3) and (x = -3). These are the points where the parabola crosses the x-axis.", "### Benefits of Using Factored Form", "1. Ease of Solving Equations\n Factored form makes solving quadratic equations straightforward. To find the roots, set each binomial equal to zero:\n [ x - 3 = 0 \implies x = 3 ]\n [ x + 3 = 0 \implies x = -3 ]\n This simple solution process wouldn’t be as direct with the expanded form (x^2 - 9 = 0), which still requires factoring or using the quadratic formula.", "2. Identifies Zeros and x-Intercepts\n The zeros of a quadratic polynomial correspond visually to the x-intercepts of its graph—a parabola crossing the x-axis. Factoring quickly reveals these values: here, (x = -3) and (x = 3) are the x-intercepts.", "3. Simplifies Polynomial Analysis\n Factored form supports easier expansion, expansion of higher-degree polynomials, and analysis of function behavior. It’s foundational in algebra, calculus, and beyond.", "### How to Recognize This Factored Form", "When seeing a quadratic in the product ((x - a)(x + a)), the symmetry around zero often hints at the difference of squares. Always check if the middle term is zero—this is a signature for factoring into a difference of squares.", "---", "In summary, the factored form (\boxed{(x - 3)(x + 3)}) is not just a convenient expression—it’s a direct application of the powerful algebraic identity (a^2 - b^2). Recognizing and using this form streamlines solving, reveals key properties like roots, and strengthens your algebraic foundation. Mastering factoring empowers deeper understanding of quadratic equations and prepares you for advanced mathematical concepts. Whether reinforcing classroom learning or conquering standardized tests, knowing this form is essential for success.", "---", "Keywords: factored form, ((x - 3)(x + 3)), difference of squares, quadratic equations, solving quadratics, algebra, factoring techniques.\nMeta description:* Understand why ( (x - 3)(x + 3) ) is the factored form of (x^2 - 9), explore the difference of squares, and learn how factoring simplifies solving quadratic equations."]