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

["# 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!"]