Solving \( n(n+1) = 420 \) gives \( n^2 + n - 420 = 0 \).

["# Solving ( n(n+1) = 420 ): Discover the Quadratic Equation and Its Solution", "When faced with the seemingly simple equation ( n(n+1) = 420 ), many students and math enthusiasts wonder: how do we solve this? What initially appears to be a straightforward problem turns into a fundamental example of transforming word problems into quadratic equations—a key skill in algebra. In this article, we’ll explore how ( n(n+1) = 420 ) leads naturally to the quadratic form ( n^2 + n - 420 = 0 ), and how to solve it step by step.", "## Understanding the Equation ( n(n+1) = 420 )", "At its core, ( n(n+1) = 420 ) expresses that the product of two consecutive whole numbers equals 420. The left side simplifies to ( n^2 + n ), transforming the original equation into:", "[\nn^2 + n = 420\n]", "By subtracting 420 from both sides, we rearrange it into the standard quadratic form:", "[\nn^2 + n - 420 = 0\n]", "This quadratic equation—( n^2 + n - 420 = 0 )—is now ready for solution using familiar methods like factoring, completing the square, or the quadratic formula.", "## Solving the Quadratic: Factoring Approach", "Quadratic equations can often be factored when they have integer solutions. We want two numbers whose product is (-420) (since ( a = 1, c = -420 )) and whose sum is (1) (the coefficient of ( n )).", "Looking for factor pairs of 420 such that one is three units larger than the other (because the sum is (+1)):", "- ( 21 \ imes 20 = 420 )\n- ( 21 - 20 = 1 )", "So, we rewrite the middle term using these numbers:", "[\nn^2 + 21n + 20n - 420 = 0\n]", "Group terms:", "[\n(n^2 + 21n) + (20n - 420) = 0\n]\n[\nn(n + 21) + 20(n + 21) = 0\n]\n[\n(n + 21)(n - 20) = 0\n]", "Setting each factor to zero:", "- ( n + 21 = 0 \Rightarrow n = -21 )\n- ( n - 20 = 0 \Rightarrow n = 20 )", "Since ( n ) typically represents a count of items or discrete items, we discard ( n = -21 ) and accept:", "[\nn = 20\n]", "## Why the Quadratic Form Matters", "The ability to convert ( n(n+1) = 420 ) into ( n^2 + n - 420 = 0 ) illustrates a powerful algebraic technique: recognizing that word problems can be generalized as quadratic equations. This expansion is not just algebraic manipulation—it’s linking real-world contexts with mathematical modeling.", "Solving this quadratic yields meaningful insight: the two consecutive integers are 20 and 21, since ( 20 \ imes 21 = 420 ). This reinforces how quadratics emerge naturally from sequences and related rates in everyday problems.", "## Alternative Methods to Solve", "While factoring is effective here, other techniques also work:", "- Quadratic Formula:\n [\n n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad a=1, b=1, c=-420\n ]\n [\n n = \frac{-1 \pm \sqrt{1 + 1680}}{2} = \frac{-1 \pm \sqrt{1681}}{2} = \frac{-1 \pm 41}{2}\n ]\n So ( n = 20 ) or ( n = -21 ), same result.", "- Graphical Method: Plotting ( y = n^2 + n - 420 ) reveals the x-intercepts at ( n = 20 ) and ( n = -21 ).", "## Applications and Takeaways", "Understanding how to transform expressions like ( n(n+1) ) into quadratics empowers students to tackle a range of problems involving patterns, sales growth, and discrete mathematical modeling. Whether managing inventory, analyzing sequences, or solving mysteries like consecutive numbers summing or multiplying to a value, this method is indispensable.", "### Key Takeaways:", "- ( n(n+1) = 420 ) → simplifies to ( n^2 + n - 420 = 0 )\n- Factoring leverages pair-sum-product logic\n- Quadratic solutions reveal discreteataset relationships\n- Algebra is the bridge between expressions and real-world problems", "Whether you’re a student mastering algebra or a curious learner exploring math’s beauty, recognizing how equations evolve is the pathway to fluency.", "---", "Keywords: ( n(n+1) = 420 ), solution ( n^2 + n - 420 = 0 ), quadratic equation, factoring, algebra, concurrent numbers, quadratic formula, real-world math, solving quadratics.", "Meta Description:\nDiscover how ( n(n+1) = 420 ) transforms into the quadratic equation ( n^2 + n - 420 = 0 ), and solve it using factoring, quadratic formula, and real-world interpretation. Learn key algebraic techniques and applications."]