\[ (10 - d)(10 + d) = 100 - d^2 = 48. \]

["# Solving ((10 - d)(10 + d) = 100 - d^2 = 48): A Step-by-Step Guide to Simple Quadratic Equations", "When faced with an equation like ((10 - d)(10 + d) = 100 - d^2 = 48), solving for (d) becomes straightforward—thanks to a fundamental algebraic identity. In this SEO-optimized guide, we’ll explore how to simplify and solve this equation, why it’s important, and real-world applications to boost your algebra skills.", "---", "## Understanding the Equation: ((10 - d)(10 + d) = 100 - d^2 = 48)", "The expression ((10 - d)(10 + d)) follows the difference of squares formula:\n[\n(a - b)(a + b) = a^2 - b^2\n]\nApplying this here:\n[\n(10 - d)(10 + d) = 10^2 - d^2 = 100 - d^2\n]", "So the equation becomes:\n[\n100 - d^2 = 48\n]", "This is a linear equation in disguise—perfect for quick solving.", "---", "## Step-by-Step Solution", "### Step 1: Isolate the variable term\nStart with:\n[\n100 - d^2 = 48\n]", "### Step 2: Subtract 48 from both sides\n[\n100 - d^2 - 48 = 0 \quad \Rightarrow \quad 52 - d^2 = 0\n]", "### Step 3: Rearrange to standard quadratic form\n[\n-d^2 + 52 = 0 \quad \Rightarrow \quad d^2 = 52\n]", "### Step 4: Take square roots\n[\nd = \pm \sqrt{52}\n]", "### Step 5: Simplify the radical\n[\n\sqrt{52} = \sqrt{4 \ imes 13} = 2\sqrt{13}\n]", "---", "## Final Answer", "[\nd = \pm 2\sqrt{13}\n]", "---", "## Why This Equation Matters: Real-World Applications", "Equations based on the difference of squares appear in physics (e.g., modeling motion), engineering (calculating areas), and finance (determining break-even points). Understanding how to simplify ((a - b)(a + b) = a^2 - b^2) helps solve these practically every day.", "---", "## Key Takeaways for Students and Learners", "- Recognize the difference of squares pattern: ((a - b)(a + b) = a^2 - b^2).\n- Always isolate (d^2) when solving linear forms derived from this identity.\n- Use radical simplification for exact answers—especially with irrational square roots.\n- Practice identifying equivalent forms to solve complex algebraic problems faster.", "---", "### Boost Your Algebra Skills Today\nUnderstanding equations like ((10 - d)(10 + d) = 48) sharpens your problem-solving toolkit. Explore more free algebra resources, practice daily, and master the power of pattern recognition in equations.", "---", "Keywords: ((10 - d)(10 + d) = 100 - d^2 = 48, difference of squares, solving quadratic equations, algebraic identities, step-by-step solution, simplify radical expressions, linearize algebra, real-world math applications.", "---", "By mastering this equation, you’re not just solving for (d)—you’re building a strong foundation for advanced math and real-life problem solving. Start practicing!"]