Factoring: \( (x+14)(x-12) = 0 \) → \( x = 12 \) (positive).
["# Understanding Factoring: Solving ( (x+14)(x-12) = 0 ) for the Positive Solution ( x = 12 )", "Factoring is a fundamental algebraic technique used to solve polynomial equations by breaking them down into simpler, multiplicative components. One classic example is solving the equation:", "[\n(x + 14)(x - 12) = 0\n]", "This equation exemplifies how factoring helps identify the values of ( x ) that make the entire expression zero—thanks to the Zero Product Property, which states that if a product of factors equals zero, then at least one of the factors must be zero.", "---", "## Why Factoring Matters in Solving Equations", "In algebra, polynomials like ( P(x) = a(x - r)(x - s) ) have roots at ( x = r ) and ( x = s ). Factoring allows us to rewrite higher-degree polynomials in factored form, revealing their roots directly and clearly.", "---", "## Step-by-Step: Solving ( (x + 14)(x - 12) = 0 )", "### Step 1: Recall the Zero Product Property", "If the product of two factors is zero:", "[\nA(x) \cdot B(x) = 0 \quad \Rightarrow \quad A(x) = 0 \quad \ ext{or} \quad B(x) = 0\n]", "---", "### Step 2: Set Each Factor Equal to Zero", "Given:\n[\n(x + 14)(x - 12) = 0\n]", "Set each factor equal to zero:", "1. ( x + 14 = 0 )\n Solve:\n [\n x = -14\n ]", "2. ( x - 12 = 0 )\n Solve:\n [\n x = 12\n ]", "---", "### Step 3: Identify the Positive Solution", "From the two solutions ( x = -14 ) and ( x = 12 ), only ( x = 12 ) is positive.", "---", "## Real-World Significance of the Positive Root", "In many real-world applications—such as economics, engineering, or physics—the sign of a solution reflects physical meaning. For example, if ( x ) represents a measurable quantity like time, distance, or cost, only positive values may be valid. Here, ( x = 12 ) could represent a positive time point, length, or other positive dimension—making it the meaningful solution in context.", "---", "## Summary", "- Factoring converts equations into products equal to zero.\n- Applying the Zero Product Property yields remaining factors.\n- The equation ( (x+14)(x-12) = 0 ) yields solutions ( x = -14 ) and ( x = 12 ).\n- Among them, ( x = 12 ) is the positive solution.", "---", "## Key Takeaway", "When solving ( (x + a)(x + b) = 0 ), factoring gives roots ( x = -a ) and ( x = -b ). Always evaluate which solution makes physical sense—especially when only positive values are meaningful. Mastering this skill enhances algebraic proficiency and problem-solving across disciplines.", "---", "## Related Keywords for SEO Optimization\nfactoring quadratics, solve (x+14)(x-12) = 0, positive root of equation, algebraic solution technique, quadratic equation factors, zero product property application, math homework help factoring, algebra root finding, positive solution algebra, factoring cellulose: x + 14 times x minus 12 equal zero, find x from (x+14)(x−12)=0, step-by-step factoring equation, why is x = 12 the solution to (x+14)(x−12) = 0", "---", "By understanding factoring through clear examples like this, learners develop a strong foundation in elementary algebra—essential for advanced math and real-life applications."]
