Simplify the expression: \( rac{(x - 3)(x + 3)}{x - 3} = x + 3 \) for \( x
["Simplify the Expression: ( \frac{(x - 3)(x + 3)}{x - 3} = x + 3 ), for ( x <br/>\ne 3 )", "Understanding how to simplify algebraic expressions is essential in algebra, and one of the most common examples is simplifying a rational expression involving a common factor in the numerator and denominator. Here, we analyze and simplify:", "[\n\frac{(x - 3)(x + 3)}{x - 3} = x + 3, \quad \ ext{for } x <br/>\ne 3\n]", "### What the Expression Represents", "The left-hand side of the equation is a fraction where the numerator is ((x - 3)(x + 3)), which is a product of a binomial and its difference — a recognizable form of the difference of squares:", "[\n(x - 3)(x + 3) = x^2 - 9\n]", "The denominator is simply (x - 3). So the expression becomes:", "[\n\frac{x^2 - 9}{x - 3}\n]", "However, rather than leaving it in this form, we look for cancellation opportunities — but only when the denominator is not zero.", "### Why the Simplification Works", "The key insight is that ((x - 3)) appears in both the numerator and the denominator. As long as (x <br/>\ne 3) (which prevents division by zero), this factor can be canceled:", "[\n\frac{(x - 3)(x + 3)}{x - 3} = x + 3\n]", "This simplification is valid under the condition (x <br/>\ne 3), because when (x = 3), the original expression becomes undefined — the denominator is zero, and division by zero is forbidden in algebra.", "### Step-by-Step Simplification", "1. Factor the numerator:\n Recognize ((x - 3)(x + 3)) as a difference of squares:\n [\n (x - 3)(x + 3) = x^2 - 9\n ]", "2. Write the expression:\n [\n \frac{x^2 - 9}{x - 3}\n ]", "3. Cancel common factor (when (x <br/>\ne 3)):\n Since (x - 3) appears in both numerator and denominator, cancel it:\n [\n \frac{x^2 - 9}{x - 3} = x + 3\n ]", "4. State the restriction:\n The simplification holds only for (x <br/>\ne 3), because at (x = 3), the original denominator is zero.", "### Practical Implications", "This simplification is valuable in solving equations, analyzing functions, and reducing rational expressions to more manageable forms. For instance, if you encounter:", "[\n\frac{(x - 3)(x + 3)}{x - 3} = 5\n]", "Replacing the left side with (x + 3) (for (x <br/>\ne 3)) transforms the equation into:", "[\nx + 3 = 5 \quad \Rightarrow \quad x = 2\n]", "You must always remember that (x = 3) is excluded — otherwise, the expression is undefined.", "### Summary", "The equivalent simplified expression is:", "[\n\frac{(x - 3)(x + 3)}{x - 3} = x + 3, \quad \ ext{for } x <br/>\ne 3\n]", "This demonstrates how recognizing common factors allows algebraic expressions to be simplified cleanly — with clear care given to domain restrictions.", "For further practice, explore similar rational expressions and practice identifying valid simplifications and excluded values.", "---", "Keywords: simplify algebraic expression, simplify ( \frac{(x - 3)(x + 3)}{x - 3} = x + 3 ), restrict domain x ≠ 3, algebraic simplification, difference of squares, rational expressions."]
