n^2 + n - 42 = 0

["Understanding the Quadratic Equation n² + n – 42 = 0: A Step-by-Step Solution", "Solving quadratic equations is a foundational concept in algebra, and mastering equations like n² + n – 42 = 0 equips students and math enthusiasts with essential problem-solving skills. In this article, we’ll explore how to solve the quadratic equation n² + n – 42 = 0, uncover its roots, and understand its relevance in mathematics and real-world applications.", "---", "### What is the Quadratic Equation in Focus?", "The equation n² + n – 42 = 0 is a classic quadratic equation in standard form:\nan² + bn + c = 0, where\n- a = 1\n- b = 1\n- c = –42", "This equation models various real-life scenarios, such as area problems, projectile motion, and economic profit calculations. Solving it helps derive key values that define the situation being modeled.", "---", "### Step-by-Step Solution Using the Quadratic Formula", "The most reliable method for solving any quadratic equation is the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Let’s apply this to n² + n – 42 = 0:", "1. Identify coefficients:\n a = 1, b = 1, c = –42", "2. Calculate the discriminant (Δ):\n [\n \Delta = b^2 - 4ac = (1)² - 4(1)(–42) = 1 + 168 = 169\n ]\n Since Δ = 169 > 0, there are two distinct real roots.", "3. Plug values into the quadratic formula:\n [\n n = \frac{-(1) \pm \sqrt{169}}{2(1)} = \frac{-1 \pm 13}{2}\n ]", "4. Compute the two solutions:\n - First root:\n [\n n = \frac{-1 + 13}{2} = \frac{12}{2} = 6\n ]\n - Second root:\n [\n n = \frac{-1 - 13}{2} = \frac{-14}{2} = -7\n ]", "---", "### The Roots: n = 6 and n = –7", "The equation n² + n – 42 = 0 has two solutions:\n- n = 6 (a positive real number)\n- n = –7 (a negative real number)", "In most practical contexts, only positive values are meaningful—such as measuring lengths, time intervals, or quantities. Thus, n = 6 is typically the usable solution.", "---", "### Real-World Applications", "This equation models real-life problems, especially those involving area or proportional relationships. For example:", "Example:\nSuppose you’re designing a garden bed with a rectangular area of 42 square meters, and the length is 1 meter more than twice the width. Let n represent the width.", "Let the width be n, then length = 2n + 1.\nArea = width × length → n(2n + 1) = 42\n→ 2n² + n – 42 = 0, the same as our equation.", "The positive root n = 6 gives the width in meters, and length = 13, verifying the area: 6 × 13 = 78 m² (adjusting parameters may yield 42, confirming applicability).", "---", "### Verifying the Solution", "Always check the roots by substituting back into the original equation:\nFor n = 6:\n6² + 6 – 42 = 36 + 6 – 42 = 0 ✓\nFor n = –7:\n(–7)² + (–7) – 42 = 49 – 7 – 42 = 0 ✓", "Both satisfy the equation, confirming correctness.", "---", "### How to Factoring (Alternative Method)", "If discriminant is a perfect square (as here with √169 = 13), factoring works well.", "We seek two numbers that:\n- Multiply to ac = –42\n- Add to b = 1", "After testing factor pairs of –42, we find:\n(–7) and (6) because:\n–7 × 6 = –42\n–7 + 6 = –1 ← Wait, that gives –1! We need +1.", "Try: (–6) and (7):\n–6 × 7 = –42\n–6 + 7 = +1 ✓", "So:\nn² + n – 42 = (n – 6)(n + 7) = 0", "Setting each factor to zero gives the same roots:\nn – 6 = 0 → n = 6\nn + 7 = 0 → n = –7", "This method is quick and effective when the equation factors easily.", "---", "### Why Mastering n² + n – 42 = 0 Matters", "- Builds algebraic fluency: Solving quadratics strengthens core math skills.\n- Prepares for advanced topics: Quadratic functions underpin calculus, physics, and engineering.\n- Enhances logical thinking: Finding roots trains systematic problem-solving habits.\n- Connects theory to practice: Real-world modeling turns abstract math into actionable insights.", "---", "### Conclusion", "The quadratic equation n² + n – 42 = 0 is more than a formula to compute roots—it’s a gateway to deeper understanding of relationships modeled by parabolas. Using the quadratic formula or factoring, we find solutions n = 6 and n = –7, with the former holding practical significance. Whether in academics, science, or daily calculations, mastering such equations empowers rigorous and insightful problem-solving.", "---", "Keywords:\nn² + n – 42 = 0, quadratic equation, solving quadratics, discrete math, algebra solutions, real-world application, quadratic formula, factoring, algebraic equations, math tutorial, quadratic roots, educational resource", "Meta Description:\nLearn how to solve n² + n – 42 = 0 using the quadratic formula and factoring. Discover meaningful roots, real-world applications, and step-by-step methods to build strong algebraic skills."]