Solve quadratic: \( n^2 + n - 420 = 0 \)

["# Solve the Quadratic Equation ( n^2 + n - 420 = 0 ): Step-by-Step Guide", "Quadratic equations are fundamental in algebra and appear widely in mathematics, physics, and engineering. One classic example is solving the equation:", "[\nn^2 + n - 420 = 0\n]", "This article walks you through solving this quadratic equation using multiple proven methods, making it easy to understand how to find real and rational solutions. Whether you're a student or a math enthusiast, mastering this process will enhance your problem-solving skills.", "---", "## Why Solve Quadratic Equations?", "Quadratic equations are key to modeling real-world situations such as projectile motion, optimization problems, and geometry. Understanding how to solve them opens doors to deeper mathematical concepts like factoring, completing the square, and using the quadratic formula.", "---", "## Step 1: Identify the Standard Form", "The equation ( n^2 + n - 420 = 0 ) is already in standard quadratic form:", "[\nan^2 + bn + c = 0\n]", "Where:\n- ( a = 1 )\n- ( b = 1 )\n- ( c = -420 )", "---", "## Step 2: Choose a Solving Method", "There are three main methods to solve quadratic equations:", "1. Factoring – Factor the quadratic expression into two binomials.\n2. Quadratic Formula – Apply ( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).\n3. Completing the Square – Rewrite the equation in perfect square form.", "For this equation, factoring is the quickest and most efficient method.", "---", "## Method 1: Factoring ( n^2 + n - 420 = 0 )", "We aim to factor the quadratic expression into two binomials:", "[\n(n + m)(n + k) = n^2 + (m + k)n + mk\n]", "We need two numbers ( m ) and ( k ) such that:", "- ( m + k = 1 )\n- ( mk = -420 )", "Trials and error or systematic checking reveal that ( m = 21 ) and ( k = -20 ) satisfy these conditions because:", "[\n21 + (-20) = 1 \quad \ ext{and} \quad 21 \ imes (-20) = -420\n]", "So, we rewrite the equation as:", "[\n(n + 21)(n - 20) = 0\n]", "Set each factor equal to zero:", "[\nn + 21 = 0 \quad \Rightarrow \quad n = -21\n]\n[\nn - 20 = 0 \quad \Rightarrow \quad n = 20\n]", "---", "## Step 3: Verify the Solutions", "Plug both values back into the original equation to confirm they satisfy ( n^2 + n - 420 = 0 ).", "For ( n = 20 ):\n[\n20^2 + 20 - 420 = 400 + 20 - 420 = 0 \quad \ ext{✓}\n]", "For ( n = -21 ):\n[\n(-21)^2 + (-21) - 420 = 441 - 21 - 420 = 0 \quad \ ext{✓}\n]", "Both solutions are correct.", "---", "## Method 2: Using the Quadratic Formula", "If factoring proves difficult, the quadratic formula offers a reliable solution:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in ( a = 1 ), ( b = 1 ), ( c = -420 ):", "[\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-420)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 1680}}{2} = \frac{-1 \pm \sqrt{1681}}{2}\n]", "Since ( \sqrt{1681} = 41 ):", "[\nn = \frac{-1 \pm 41}{2}\n]", "This gives two solutions:", "[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20\n]\n[\nn = \frac{-1 - 41}{2} = \frac{-42}{2} = -21\n]", "Again, we confirm:\nSolutions: ( n = 20 ), ( n = -21 )", "---", "## Why the Solutions Make Sense", "The roots ( n = 20 ) and ( n = -21 ) represent points where the quadratic function crosses the x-axis. Since the parabola opens upward (coefficient of ( n^2 ) is positive), the function dips below the axis between these two real roots, confirming two real solutions.", "---", "## Conclusion", "Solving ( n^2 + n - 420 = 0 ) yields the solutions:", "[\n\boxed{n = -21 \quad \ ext{and} \quad n = 20}\n]", "These values are derived using factoring and the quadratic formula—two powerful tools in algebra. Understanding these methods helps simplify complex equations and builds confidence in tackling quadratic problems.", "---", "## Further Reading", "- Mastering Quadratic Equations in Real-World Applications\n- Quadratic Formula vs. Factoring: When to Use Each\n- Graphing Quadratic Functions Using Roots", "---", "Keywords: Solve quadratic equation, quadratic formula, factoring quadratic, n squared plus n minus 420, solve n² + n − 420 = 0, quadratic solutions, algebra tutorial, math problem solving."]