The expression becomes \( rac{(x - 2)(x + 2)}{x - 2} \).
["# Understanding and Simplifying the Expression: ( \frac{(x - 2)(x + 2)}{x - 2} )", "When encountering the algebraic expression\n[\n\frac{(x - 2)(x + 2)}{x - 2},\n]\nit’s common for learners to seek clarity on simplification, domain restrictions, and mathematical meaning. This article clarifies how to interpret and reduce this expression step by step.", "## What Is the Expression?", "The given expression is a rational function written in factored form:\n[\n\frac{(x - 2)(x + 2)}{x - 2}.\n]\nAt first glance, the numerator is a product of two binomials ((x - 2)(x + 2)), and the denominator is the linear term ((x - 2)).", "## Simplifying the Expression", "Because the numerator and denominator share a common factor ((x - 2)), we can simplify — but only under important conditions:", "### Step 1: Factor Cancellation\nFor ( x <br/>\neq 2 ), the factor ((x - 2)) cancels out numerator and denominator:\n[\n\frac{(x - 2)(x + 2)}{x - 2} = x + 2, \quad \ ext{provided } x <br/>\neq 2.\n]", "Note: The cancellation is valid only when ( x - 2 <br/>\ne 0 ), i.e., ( x <br/>\ne 2 ). The original expression is undefined at ( x = 2 ) due to division by zero.", "### Step 2: Domain Consideration\nThough algebraically reduced to ( x + 2 ), the expression retains a domain restriction from the original form:\n- Domain: All real numbers except ( x = 2 ).\n- At ( x = 2 ), the function is undefined.", "## Why This Simplification Matters", "- Simplifies calculations: Reducing complex fractions to linear expressions facilitates easier evaluation and manipulation.\n- Reveals behavior near discontinuities: Understanding where the simplification fails highlights discontinuities in rational functions.\n- Foundation for deeper algebra: Emphasizes domain rules, factoring, and rational expression properties.", "## Example Evaluation", "- At ( x = 3 ):\n Original:\n [\n \frac{(3 - 2)(3 + 2)}{3 - 2} = \frac{1 \cdot 5}{1} = 5\n ]\n Simplified:\n [\n 3 + 2 = 5 \quad \ ext{(Correct, for } x <br/>\ne 2\ ext{)}\n ]\n- At ( x = 2 ):\n Original is undefined. The simplified form ( x + 2 = 4 ) appears plausible but does not match the original function’s behavior at that point.", "## Final Notes", "While the expression simplifies elegantly to ( x + 2 ) for ( x <br/>\ne 2 ), always preserve the restriction ( x <br/>\ne 2 ). Understanding both the simplified form and its domain ensures accurate algebraic reasoning and avoids common errors in calculus, equation solving, and graphing rational functions.", "---", "Key Lessons Summary:\n- Cancelling common factors in fractions is valid only when denominators are non-zero.\n- Expressions like ( \frac{(x - 2)(x + 2)}{x - 2} ) simplify to ( x + 2 ), with a hole or discontinuity at ( x = 2 ).\n- Recognizing domain constraints enriches mathematical understanding and application.", "For further practice, try simplifying other rational expressions and remember: always check where the original denominator equals zero.", "---", "Keywords: ( \frac{(x - 2)(x + 2)}{x - 2} ), simplify rational expression, algebraic simplification, domain restrictions, x ≠ 2, canceling factors, algebraic expressions, function simplification, solving rational equations."]
