Simplify \( (x^2 - 9)/(x + 3) \) and state any restrictions.
["SEO Optimized Article: Simplifying ( \frac{x^2 - 9}{x + 3} ) – Step-by-Step Guide with Restrictions", "---", "# Simplify ( \frac{x^2 - 9}{x + 3} ): Step-by-Step Simplification with Key Restrictions", "When working with rational expressions like ( \frac{x^2 - 9}{x + 3} ), simplifying them is essential for solving equations, analyzing functions, and performing algebraic operations. In this article, we’ll walk you through how to simplify ( \frac{x^2 - 9}{x + 3} ), explain the algebraic reasoning, and highlight important restrictions you must consider.", "---", "## What is the Expression?", "The expression to simplify is:", "[\n\frac{x^2 - 9}{x + 3}\n]", "This is a rational function — a fraction where both numerator and denominator are polynomials. Simplifying rational expressions usually involves factoring and canceling common terms, but only when those terms are valid.", "---", "## Step 1: Factor the Numerator", "Start by factoring the numerator ( x^2 - 9 ). Recognize that this is a difference of squares, which factors as:", "[\nx^2 - 9 = (x - 3)(x + 3)\n]", "So now the expression becomes:", "[\n\frac{(x - 3)(x + 3)}{x + 3}\n]", "---", "## Step 2: Simplify the Fraction", "Next, cancel common factors in the numerator and denominator — but only when the denominator is not zero.", "Since ( x + 3 ) appears in both numerator and denominator, we can cancel it:", "[\n\frac{(x - 3)\cancel{(x + 3)}}{\cancel{x + 3}} = x - 3 \quad \ ext{(for } x <br/>\ne -3\ ext{)}\n]", "---", "## Final Simplified Form", "The simplified expression is:", "[\nx - 3, \quad \ ext{provided } x <br/>\ne -3\n]", "This means the original rational expression is equivalent to ( x - 3 ), except when ( x = -3 ), which makes the denominator zero and is undefined.", "---", "## Why Are Restrictions Important?", "While simplifying ( \frac{x^2 - 9}{x + 3} ), the key restriction is:", "( x <br/>\ne -3 )", "Why? Because division by zero is undefined in mathematics. Even though the expression simplifies neatly to ( x - 3 ) algebraically, plugging ( x = -3 ) into the original denominator yields division by zero:", "[\nx + 3 = -3 + 3 = 0 \quad \ ext{(undefined)}\n]", "Thus, ( x = -3 ) is not part of the domain, and the simplified expression must still exclude this value.", "---", "## Practical Use", "Simplifying rational expressions is crucial in algebra, calculus, and applied math. The simplified form ( x - 3 ) makes it easier to graph functions, solve equations, and perform calculus operations. However, always bear in mind the domain restriction to avoid invalid conclusions (like substituting ( x = -3 )).", "---", "## Summary", "- Simplify ( \frac{x^2 - 9}{x + 3} ) by factoring numerator and canceling common terms\n- Issue: denominator ( x + 3 <br/>\ne 0 \Rightarrow x <br/>\ne -3 )\n- Simplified expression: ( x - 3 ), valid only when ( x <br/>\ne -3 )", "---", "### Key Takeaway:", "[\n\boxed{ \frac{x^2 - 9}{x + 3} = x - 3 \quad \ ext{for } x <br/>\ne -3 }\n]", "Understanding this simplification and its domain restriction ensures you work accurately and confidently with rational expressions online and in exams.", "---", "For further reading, explore factoring techniques, domain analysis, and rational expressions in algebra.", "---", "Keywords for SEO:\n[\nsimplify ( \frac{x^2 - 9}{x + 3} ), simplify rational expressions, algebraic simplification, domain restrictions, cancel common factors, ( x <br/>\ne -3 ), step-by-step simplification, conditional simplification\n]", "Meta Description:\nLearn how to simplify ( \frac{x^2 - 9}{x + 3} ) step-by-step, including factoring, canceling common terms, and the important restriction ( x <br/>\ne -3 ) to avoid undefined expressions."]
