Solve \(n^2 + n - 420 = 0\) using the quadratic formula:
["How to Solve ( n^2 + n - 420 = 0 ) Using the Quadratic Formula", "Solving quadratic equations forms a fundamental skill in algebra, enabling students and math enthusiasts alike to understand powerful problem-solving techniques. One common quadratic equation students often encounter is:", "[\nn^2 + n - 420 = 0\n]", "This article explains how to solve this equation using the quadratic formula, provides step-by-step calculations, and highlights practical insights to help reinforce your understanding.", "---", "### Understanding the Quadratic Formula", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "The quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "gives the solutions (roots) for any quadratic equation. Here, (a), (b), and (c) are known coefficients, and the symbol (\Delta = b^2 - 4ac) determines the nature of the roots—real and distinct, real and repeated, or complex.", "---", "### Step 1: Identify coefficients from the equation", "Given:\n[\nn^2 + n - 420 = 0\n]", "Comparing with (an^2 + bn + c = 0), we identify:\n- (a = 1)\n- (b = 1)\n- (c = -420)", "---", "### Step 2: Plug coefficients into the quadratic formula", "Substitute (a = 1), (b = 1), and (c = -420) into the formula:", "[\nn = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-420)}}{2(1)}\n]", "Simplify step by step:", "[\nn = \frac{-1 \pm \sqrt{1 + 1680}}{2}\n]", "[\nn = \frac{-1 \pm \sqrt{1681}}{2}\n]", "---", "### Step 3: Calculate the square root", "Find (\sqrt{1681}):\nSince (41^2 = 1681), we know:", "[\n\sqrt{1681} = 41\n]", "---", "### Step 4: Simplify the expression", "Now substitute back:", "[\nn = \frac{-1 \pm 41}{2}\n]", "This gives two solutions:", "First solution:", "[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20\n]", "Second solution:", "[\nn = \frac{-1 - 41}{2} = \frac{-42}{2} = -21\n]", "---", "### Step 5: Final solution", "The equation ( n^2 + n - 420 = 0 ) has two real solutions:", "[\n\boxed{n = 20} \quad \ ext{and} \quad \boxed{n = -21}\n]", "---", "### Why Using the Quadratic Formula Works", "The quadratic formula delivers accurate results regardless of the complexity of (b^2 - 4ac). In this case, a positive discriminant ((1681)) means two distinct real roots. The formula efficiently handles all sign combinations to find both solutions.", "---", "### Practice Tip", "Always simplify the square root before plugging into the formula—avoiding large decimals or inaccuracies. Recognizing perfect squares, like (\sqrt{1681} = 41), speeds up solving significantly.", "---", "### When to Use the Quadratic Formula vs. Factoring", "While factoring is faster in simple cases, the quadratic formula works reliably for all quadratics—especially when coefficients are large or factoring isn’t straightforward. Practice both methods to build flexibility.", "---", "### Summary", "Solving ( n^2 + n - 420 = 0 ) using the quadratic formula gives two clear real solutions:", "[\n\boxed{n = 20} \quad \ ext{and} \quad \boxed{n = -21}\n]", "Mastering the quadratic formula empowers you to tackle any quadratic equation confidently—key for advanced math and real-world applications.", "---", "Keywords: solve (n^2 + n - 420 = 0), quadratic formula, algebra, quadratic equation solutions, discriminant, step-by-step algebra, mathematics practice, high school math, quadratic roots", "---", "Ready to solve your next quadratic equation? Strengthen your problem-solving with the quadratic formula today!"]
