Multiply both sides by 2: \( n(n+1) = 420 \).
["Multiply Both Sides by 2: Solving the Equation ( n(n+1) = 420 )", "When solving quadratic equations in algebra, one common technique is to manipulate the equation to make it easier to solve. In this article, we’ll explore a straightforward method: multiplying both sides of the equation ( n(n+1) = 420 ) by 2. This step simplifies the expression and prepares it for factoring or applying the quadratic formula—key skills for any student tackling algebraic systems.", "---", "### Understanding the Equation", "We start with:", "[\nn(n+1) = 420\n]", "This equation represents a product of two consecutive terms, ( n ) and ( n+1 ), equaling 420. Expanding the left-hand side gives:", "[\nn^2 + n = 420\n]", "However, multiplying both sides of the original equation by 2 instead allows us to avoid fully expanding right away, which can streamline the solving process. Let's see why.", "---", "### Why Multiply Both Sides by 2?", "Multiplying both sides by 2 yields:", "[\n2n(n + 1) = 840\n]", "Now the left side becomes:", "[\n2n(n + 1) = 2n^2 + 2n\n]", "This takes us straight to a quadratic form:", "[\n2n^2 + 2n - 840 = 0\n]", "This is now ready to be simplified by dividing the entire equation by 2:", "[\nn^2 + n - 420 = 0\n]", "Factoring this quadratic becomes simpler due to the simplified coefficients.", "---", "### Solve the Simplified Quadratic", "We now solve:", "[\nn^2 + n - 420 = 0\n]", "We look for two numbers that multiply to (-420) and add to (1).", "After testing factor pairs of 420, we find:", "[\n21 \ imes (-20) = -420 \quad \ ext{and} \quad 21 + (-20) = 1\n]", "Thus, we factor:", "[\n(n + 21)(n - 20) = 0\n]", "Setting each factor equal to zero:", "[\nn + 21 = 0 \quad \Rightarrow \quad n = -21\n]\n[\nn - 20 = 0 \quad \Rightarrow \quad n = 20\n]", "---", "### Verify the Solutions", "It is always wise to check solutions in the original equation:", "- For ( n = 20 ):", "[\n20(20 + 1) = 20 \ imes 21 = 420 \quad \checkmark\n]", "- For ( n = -21 ):", "[\n-21(-21 + 1) = -21 \ imes (-20) = 420 \quad \checkmark\n]", "Both values satisfy the original equation.", "---", "### The Power of Multiplying by 2 in Word Problems", "In real-world contexts, such equations model relationships where a variable and its successor interact—like population growths, financial projections, or physics problems involving consecutive integers. Multiplying by 2 offers algebraic flexibility, avoiding overflow while maintaining equivalence. This technique enhances problem-solving efficiency and strengthens conceptual understanding.", "---", "### Summary", "- Multiplying both sides of ( n(n+1) = 420 ) by 2 simplifies the equation to ( 2n(n+1) = 840 ).\n- This leads to the quadratic ( n^2 + n - 420 = 0 ), easier to factor.\n- The solutions are ( n = 20 ) and ( n = -21 ), both valid.\n- Using this method streamlines algebra and improves clarity in solving quadratic equations.", "---", "### SEO Keywords Focused:", "- Multiply both sides by 2\n- Solve ( n(n+1) = 420 )\n- Algebra quadratic solving\n- Factor quadratic equation\n- Solve ( 2n(n+1) = 840 )\n- Find integer solutions to quadratic equations", "---", "Mastering techniques like multiplying both sides by 2 helps build algebraic fluency. This approach transforms complex equations into simpler, solvable forms—your key to confidently solving equations in school and beyond."]
