["# Solving ( n(n+1) = 420 ): A Step-by-Step Guide", "If you're tackling the equation ( n(n+1) = 420 ), you're not just solving for ( n )—you're exploring a classic example of a quadratic diophantine equation. Understanding how to solve such equations is valuable in algebra, number theory, and competitive math. In this article, we break down how to solve ( n(n+1) = 420 ), interpret its solutions, and explore why this equation matters.", "---", "## What Does the Equation ( n(n+1) = 420 ) Mean?", "The expression ( n(n+1) ) represents the product of two consecutive integers. Mathematically, this is equivalent to:", "[
\nn^2 + n = 420
\n]", "Rewriting it as a standard quadratic equation:", "[
\nn^2 + n - 420 = 0
\n]", "This is a downward-opening parabola in quadratic form ( ax^2 + bx + c = 0 ), with ( a = 1 ), ( b = 1 ), and ( c = -420 ).", "---", "## How to Solve the Equation", "There are two main methods: factoring and the quadratic formula. Let’s explore both.", "### Method 1: Factoring by Inspection", "We want two consecutive integers ( n ) and ( n+1 ) whose product is 420.", "Check nearby perfect squares around 420:", "- ( 20^2 = 400 ), close to 420
\n- ( 21^2 = 441 ), a bit higher", "Try ( n = 20 ):
\n( 20 \ imes 21 = 420 ) ✅", "So one solution is:", "[
\nn = 20
\n]", "Because ( n ) and ( n+1 ) are 20 and 21, the pair ( (20, 21) ) multiplies to 420.", "### Method 2: Using the Quadratic Formula", "From ( n^2 + n - 420 = 0 ), apply the quadratic formula:", "[
\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Substitute ( 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 ), we get:", "[
\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20
\n\quad \ ext{or} \quad
\nn = \frac{-1 - 41}{2} = \frac{-42}{2} = -21
\n]", "So the two solutions are:", "[
\nn = 20 \quad \ ext{and} \quad n = -21
\n]", "---", "## Which Solution Makes Sense?", "Since ( n(n+1) = 420 ) implies two consecutive integers, and generally in contexts like counting or discrete problems, ( n ) is expected to be a positive integer, the valid solution is:", "[
\nn = 20
\n]", "The negative solution ( n = -21 ) yields ((-21)(-20) = 420), but is less commonly relevant unless specified.", "---", "## Why ( n(n+1) = 420 ) Matters", "This equation is a beautiful example of how consecutive integers can model real-world scenarios—such as combinations or area problems. It also introduces key algebraic concepts:", "- Quadratic equations: Foundation for algebra, used in physics, economics, and engineering.
\n- Integer solutions: Important in cryptography and number theory.
\n- Factoring vs. quadratics: Teaches multiple approaches to solving equations.", "Solving ( n(n+1) = 420 ) helps sharpen problem-solving skills applicable far beyond this specific problem.", "---", "## Final Answer", "The primary solution to ( n(n+1) = 420 ) is:", "[
\n\boxed{n = 20}
\n]", "Thus, ( 20 \ imes 21 = 420 ), confirming the equation holds true.", "---", "## Related Searches (SEO Keywords)", "- Solve ( n(n+1) = 420 <br/>\n- Find integer solutions to ( n(n+1) = 420
\n- Quadratic equations with consecutive integers
\n- Solve ( n^2 + n - 420 = 0
\n- Algebraic problems for high school students
\n- Successive integers product equation", "---", "## Summary", "The equation ( n(n+1) = 420 ) expands your understanding of quadratics and integer relationships. Solving it yields ( n = 20 ) (and ( n = -21 )), highlighting two consecutive integers whose product is 420. Whether tackling homework, math competitions, or real-world modeling, mastering such equations strengthens algebraic intuition.", "If you enjoyed this explanation, share it with fellow learners—and remember: every equation tells a story waiting to be solved."]
