We need to factor the quadratic expression \(3x^2 + 21x + 18\).
["# How to Factor the Quadratic Expression (3x^2 + 21x + 18)", "Factoring quadratic expressions is a fundamental skill in algebra that helps simplify equations and solve quadratic problems. One commonly encountered expression is (3x^2 + 21x + 18). Factoring this quadratic expression efficiently not only boosts mathematical fluency but also prepares students and learners for more advanced topics like solving quadratic equations and graphing parabolas.", "## Why Factor Quadratic Expressions?", "Before diving into the solution, it’s important to understand why factoring matters:", "- Simplifies Expressions: Factored form highlights roots and key features of the quadratic.\n- Solves Equations Easily: Factoring allows direct application of the zero-product property.\n- Enhances Conceptual Understanding: Recognizing patterns like GCF, perfect squares, and difference of squares strengthens algebraic logic.", "## Step-by-Step Factorization of (3x^2 + 21x + 18)", "### Step 1: Factor Out the Greatest Common Factor (GCF)", "The expression (3x^2 + 21x + 18) has coefficients 3, 21, and 18. The GCF of these numbers is 3. Factoring out 3 gives:", "[\n3(x^2 + 7x + 6)\n]", "Now, we focus on factoring the quadratic inside the parentheses: (x^2 + 7x + 6).", "### Step 2: Identify Two Numbers That Multiply to (ac) and Add to (b)", "For the quadratic (x^2 + 7x + 6):", "- (a = 1), (b = 7), (c = 6)\n- We need two numbers that multiply to (1 \ imes 6 = 6) and add to (7)", "The numbers 1 and 6 satisfy both conditions:", "[\n1 \ imes 6 = 6 \quad \ ext{and} \quad 1 + 6 = 7\n]", "### Step 3: Rewrite the Middle Term Using the Two Numbers", "Break the middle term (7x) into (1x + 6x):", "[\nx^2 + 7x + 6 = x^2 + 1x + 6x + 6\n]", "### Step 4: Factor by Grouping", "Group terms to factor by grouping:", "[\n(x^2 + 1x) + (6x + 6)\n]", "Factor each group:", "[\nx(x + 1) + 6(x + 1)\n]", "Now, factor out the common binomial ((x + 1)):", "[\n(x + 1)(x + 6)\n]", "### Step 5: Include the GCF", "Recall we factored out 3 earlier:", "[\n3(x + 1)(x + 6)\n]", "## Final Answer", "The fully factored form of (3x^2 + 21x + 18) is:", "[\n\boxed{3(x + 1)(x + 6)}\n]", "## Using the Factored Form", "To verify, expand (3(x + 1)(x + 6)):", "[\n(x + 1)(x + 6) = x^2 + 6x + x + 6 = x^2 + 7x + 6\n]\n[\n3(x^2 + 7x + 6) = 3x^2 + 21x + 18\n]", "The original expression is recovered, confirming the factorization is correct.", "## Online Tools & Practice Tips", "For additional practice:", "- Use factor calculators to verify your work.\n- Apply the AC Method (multiply (a) and (c), find factors) consistently.\n- Practice with different quadratics to recognize patterns smoothly.", "---", "Summary: Factoring (3x^2 + 21x + 18) begins by extracting the GCF (3), then factoring the resulting trinomial—(x^2 + 7x + 6)—by finding two numbers that multiply to 6 and add to 7 (namely 1 and 6). Grouping and factoring by common terms yield the final factored form: (3(x + 1)(x + 6)). This process strengthens algebra skills essential for solving quadratics and graphing functions.", "---", "Keywords: factor quadratic expression, factor (3x^2 + 21x + 18), step-by-step factoring, algebraic manipulation, finding GCF, factoring by grouping, algebraic skills, solve quadratic equations, simplify expressions."]
