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

["Solve the Quadratic Equation: ( n^2 + n - 420 = 0 ) – Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, essential for students, educators, and professionals alike. One commonly encountered quadratic equation is ( n^2 + n - 420 = 0 ). In this article, we’ll walk through the process of solving this equation step by step, explaining key concepts and providing practical insights for anyone tackling similar problems.", "---", "### What is a Quadratic Equation?", "A quadratic equation is any equation of the form:", "[\nax^2 + bx + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\ne 0 ). The equation ( n^2 + n - 420 = 0 ) fits this pattern with:", "- ( a = 1 )\n- ( b = 1 )\n- ( c = -420 )", "---", "### Step-by-Step Solution: Solving ( n^2 + n - 420 = 0 )", "#### Step 1: Factor the quadratic expression (if possible)", "We begin by factoring the left-hand side. Since ( a = 1 ), we look for two numbers that:", "- Multiply to ( c = -420 )\n- Add to ( b = 1 )", "Finding such numbers can take some trial, but through systematic checking, we find:", "[\n(n + 21)(n - 20) = 0\n]", "Let’s verify the factorization:", "[\n(n + 21)(n - 20) = n^2 - 20n + 21n - 420 = n^2 + n - 420\n]", "✅ Confirmed! The equation factors correctly.", "#### Step 2: Apply the Zero Product Property", "Set each factor equal to zero:", "[\nn + 21 = 0 \quad \ ext{or} \quad n - 20 = 0\n]", "Solving these gives:", "[\nn = -21 \quad \ ext{or} \quad n = 20\n]", "---", "### Why These Solutions Matter", "The solutions ( n = -21 ) and ( n = 20 ) are the roots of the equation. They represent the values of ( n ) at which the quadratic expression equals zero. Graphically, these are the points where the parabola ( y = n^2 + n - 420 ) crosses the ( n )-axis.", "---", "### Alternative Method: The Quadratic Formula", "Although factoring works well here, not all quadratics factor easily. An alternative is using the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plugging 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 ), we get:", "[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20\n]\n[\nn = \frac{-1 - 41}{2} = \frac{-42}{2} = -21\n]", "Same results — efficient and accurate!", "---", "### Tips for Solving Quadratics Quickly", "- Try factoring first — if integers work, it’s faster than formulas.\n- Use the quadratic formula when factoring isn’t obvious.\n- Check solutions by plugging them back into the original equation to confirm.\n- Recognize patterns like difference of squares or perfect trinomials.", "---", "### Real-World Applications", "Quadratic equations model real-life scenarios such as projectile motion, profit optimization, and geometric problems. Understanding how to solve them empowers students and professionals to model and solve practical problems effectively.", "---", "### Final Answer", "The solutions to the quadratic equation ( n^2 + n - 420 = 0 ) are:", "[\n\boxed{n = -21 \quad \ ext{and} \quad n = 20}\n]", "---", "### Summary", "Solving ( n^2 + n - 420 = 0 ) involves factoring, applying the quadratic formula, or using other algebraic methods. Factoring reveals quick solutions, while the quadratic formula ensures accuracy. Mastering these techniques is key to success in algebra and beyond.", "---", "Blog keywords: solve quadratic equation, quadratic formula, factoring quadratic equations, solve ( n^2 + n - 420 = 0 ), step-by-step algebra, solving for n, algebraic methods, quadratic solutions."]