["# Solving the Quadratic Equation: 2n² + 2n – 84 = 0", "Quadratic equations are fundamental in algebra and arise in various real-world applications such as physics, engineering, and economics. One such equation is 2n² + 2n – 84 = 0. In this comprehensive guide, we’ll explore how to solve this quadratic equation step-by-step, interpret its solutions, and understand its significance in mathematical modeling.", "---", "## What is the Equation?", "The equation 2n² + 2n – 84 = 0 is a standard quadratic equation in the form:", "[
\nan^2 + bn + c = 0
\n]", "where:
\n- ( a = 2 )
\n- ( b = 2 )
\n- ( c = -84 )", "Solving quadratic equations unlocks key insights into parabolas, root behavior, and applications across scientific fields.", "---", "## Step-by-Step Solution Using the Quadratic Formula", "The most reliable method to solve any quadratic equation is the quadratic formula:", "[
\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "### Step 1: Identify coefficients", "Given:
\n( a = 2 ), ( b = 2 ), ( c = -84 )", "### Step 2: Compute the discriminant ( D )", "The discriminant determines the nature of the roots:", "[
\nD = b^2 - 4ac = (2)^2 - 4(2)(-84) = 4 + 672 = 676
\n]", "Since ( D = 676 > 0 ), the equation has two distinct real roots.", "### Step 3: Plug into the quadratic formula", "[
\nn = \frac{-2 \pm \sqrt{676}}{2 \ imes 2} = \frac{-2 \pm 26}{4}
\n]", "### Step 4: Solve for both roots", "[
\nn_1 = \frac{-2 + 26}{4} = \frac{24}{4} = 6
\n]
\n[
\nn_2 = \frac{-2 - 26}{4} = \frac{-28}{4} = -7
\n]", "---", "## Interpretation of the Roots", "The solutions:", "- n = 6
\n- n = -7", "These values represent the roots of the equation. In practical modeling:", "- n = 6 could represent a physical quantity like time, distance, or concentration where the system satisfies the condition.
\n- n = -7, while mathematically valid, might be interpreted as a negative or imaginary context constraint, depending on application (e.g., only positive n acceptable in real-world problems).", "---", "## Alternative Methods: Factoring and Verification", "Although the quadratic formula is robust, let’s briefly explore factoring for insight:", "Start with:
\n( 2n^2 + 2n - 84 = 0 )", "Divide entire equation by 2:", "( n^2 + n - 42 = 0 )", "Factor:
\nFind two numbers whose product is -42 and sum is 1 → 7 and -6", "[
\n(n + 7)(n - 6) = 0
\n]", "So,
\n( n = -7 ) or ( n = 6 )—matching our earlier result.", "---", "## Why This Equation Matters", "Quadratic equations like 2n² + 2n – 84 = 0 appear in:
\n- Projectile motion modeling (where ( n ) may represent time intervals)
\n- Optimization problems in economics
\n- Geometry problems involving areas and distances", "Understanding how to solve and interpret such equations builds a foundation for advanced mathematics and real-world problem-solving.", "---", "## Final Summary", "The equation 2n² + 2n – 84 = 0 has two real roots:
\n[
\n\boxed{n = 6} \quad \ ext{and} \quad \boxed{n = -7}
\n]", "Using the quadratic formula or factoring confirms these solutions efficiently. Remember to interpret roots in context—positively versus negatively—especially in applied mathematics.", "---", "## Want to Learn More?", "- Explore how changing coefficients affects graph shape and roots.
\n- Apply the quadratic formula to real-life scenarios like business revenue models.
\n- Dive into completing the square as an alternative solution method.", "Mastering quadratic equations opens doors—try practicing with different coefficients to strengthen your skills!", "---", "Keywords for SEO:
\n2n² + 2n – 84 = 0, solving quadratic equations, quadratic formula, real roots, algebra homework help, quadratic discriminant, environment of quadratic equations, step-by-step quadratic solution, factoring quadratics, root interpretation.", "---", "Stay tuned for more in-depth algebra tutorials that bridge theory and real-world application!"]
