Solution: Simplify the left side: $ rac{(x-3)(x+3)}{x-3} = x + 3 $, provided $ x
["Simplify the Equation: $ \frac{(x-3)(x+3)}{x-3} = x + 3 $ – The Simplified Solution", "When solving algebraic equations, simplifying expressions efficiently is key to clearer understanding and faster problem solving. One common challenge students face is simplifying rational expressions involving fractions. Consider the equation:", "$$\n\frac{(x-3)(x+3)}{x-3} = x + 3, \quad \ ext{provided } x <br/>\ne 3\n$$", "At first glance, this equation may seem tricky due to the parentheses and the denominator. However, simplifying this expression step-by-step reveals its elegance and logical flow.", "### Understanding the Expression", "The left-hand side, $ \frac{(x-3)(x+3)}{x-3} $, involves a rational expression — a fraction where both numerator and denominator are polynomials. The numerator, $ (x-3)(x+3) $, is a product of two binomials. This structure follows the difference of squares formula, which states:", "$$\na^2 - b^2 = (a - b)(a + b)\n$$", "Here, $ x-3 $ and $ x+3 $ fit this pattern with $ a = x $, $ b = 3 $. Therefore:", "$$\n(x - 3)(x + 3) = x^2 - 9\n$$", "So the original expression becomes:", "$$\n\frac{x^2 - 9}{x - 3}\n$$", "### Simplifying for Restricted Domain", "However, direct simplification depends critically on the value of $ x $. The denominator $ x - 3 $ must not be zero, so we exclude $ x = 3 $. With $ x <br/>\ne 3 $, the expression is valid, and we can simplify:", "$$\n\frac{(x-3)(x+3)}{x-3} = x + 3\n$$", "Note: Even though $ x - 3 $ cancels, it’s essential to remember the restriction $ x <br/>\ne 3 $, because division by zero is undefined.", "### Final Simplified Form", "Thus, the simplified equation is:", "$$\nx + 3 = x + 3, \quad \ ext{for } x <br/>\ne 3\n$$", "### Why This Matters", "Simplifying such expressions is more than just solving algebra — it strengthens conceptual understanding of functions, domains, and algebraic identities. Recognizing the difference of squares helps in simplifying complex fractions, while acknowledging domain restrictions prevents invalid solutions.", "Key Takeaways:", "- The simplification $ \frac{(x-3)(x+3)}{x-3} = x + 3 $ is valid only when $ x <br/>\ne 3 $.\n- The expression reduces cleanly using the difference of squares: $ x^2 - 9 = (x-3)(x+3) $.\n- Understanding domain restrictions ensures mathematical accuracy.", "By mastering this simplification, learners gain confidence in manipulating rational expressions — a fundamental skill across higher-level math, science, and engineering applications.", "---", "If you found this simplification helpful, explore more about algebraic expressions and domain analysis — essential building blocks for advanced problem solving!"]
