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

["# Solve the Quadratic: ( n^2 + n - 420 = 0 )", "Quadratic equations are essential in algebra and appear frequently in science, engineering, and economics. One common challenge is solving equations of the form ( n^2 + bn + c = 0 ), like ( n^2 + n - 420 = 0 ). In this article, we’ll walk through how to solve ( n^2 + n - 420 = 0 ) using multiple methods, making it easy to understand and apply in various real-world problems.", "## Why Solve Quadratic Equations?", "Quadratic equations model many real-life scenarios such as projectile motion, optimization problems, and financial calculations. Solving ( n^2 + n - 420 = 0 ) specifically can represent time intervals, dimensions, or disparities of positive quantities.", "---", "## Solving ( n^2 + n - 420 = 0 ) Using Factoring", "The equation ( n^2 + n - 420 = 0 ) is factorable because we look for two numbers that multiply to (-420) and add to (+1).", "### Step 1: Identify two numbers\nWe need two numbers ( m ) and ( k ) such that:\n- ( m \ imes k = -420 )\n- ( m + k = 1 )", "After testing factor pairs of 420, we find:\n( m = 21 ), ( k = -20 ), since ( 21 \ imes (-20) = -420 ) and ( 21 + (-20) = 1 ).", "### Step 2: Rewrite and factor\nRewrite:\n[ n^2 + 21n - 20n - 420 = 0 ]\nGroup terms:\n[ (n^2 + 21n) - (20n + 420) = 0 ]\nFactor:\n[ n(n + 21) - 20(n + 21) = 0 ]\nNow factor out ( (n + 21) ):\n[ (n + 21)(n - 20) = 0 ]", "### Step 3: Solve for ( n )\nSet each factor to zero:\n- ( n + 21 = 0 ) → ( n = -21 )\n- ( n - 20 = 0 ) → ( n = 20 )", "---", "## Interpret the Solution: ( n = -21 ) or ( n = 20 )", "Since ( n ) often represents a physical quantity like time or length, only the positive solution is meaningful. Therefore,\n( n = 20 ) is the valid solution.", "For example, if ( n ) represents the number of days, this could describe when a growing process reaches a significant milestone.", "---", "## Alternative Method: Using the Quadratic Formula", "The quadratic formula:\n[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]\nFor ( n^2 + n - 420 = 0 ), coefficients are:\n- ( a = 1 ), ( b = 1 ), ( c = -420 )", "### Step 1: Compute the discriminant\n[ \Delta = b^2 - 4ac = 1^2 - 4(1)(-420) = 1 + 1680 = 1681 ]", "### Step 2: Take the square root\n[ \sqrt{1681} = 41 ] (since ( 41^2 = 1681 ))", "### Step 3: Apply the formula\n[ n = \frac{-1 \pm 41}{2(1)} = \frac{-1 \pm 41}{2} ]\nSo:\n- ( n = \frac{-1 + 41}{2} = \frac{40}{2} = 20 )\n- ( n = \frac{-1 - 41}{2} = \frac{-42}{2} = -21 )", "Again, discriminant-based solving confirms the same positive solution ( n = 20 ).", "---", "## Summary", "Solving ( n^2 + n - 420 = 0 ) yields:\n- Factored form: ( (n + 21)(n - 20) = 0 )\n- Solutions: ( n = -21 ) and ( n = 20 )\n- Primary solution: ( n = 20 ) (valid in most contexts)\n- Alternate method via quadratic formula validates the result", "### Pro Tip:\nWhen solving quadratic equations, always check if both solutions make sense in the problem context—positive integers are often expected.", "---", "## FAQs", "Q: Why does factoring work?\nFactoring leverages the structure of multiplication and addition to break the equation into simpler components.", "Q: Can I always factor a quadratic?\nNot all quadratics factor nicely; for those, the quadratic formula delivers accurate solutions.", "Q: How do I use quadratic equations in real life?\nFrom physics (motion) to business (profit) and architecture (design), quadratics model changes and optimize outcomes.", "---", "If you're studying algebra, mastering quadratic equations like ( n^2 + n - 420 = 0 ) strengthens your problem-solving foundation. Practice factoring, using formulas, and interpreting results—this skill will serve you across STEM fields!"]