Multiply both sides by 2: \( n(n + 1) = 10100 \).
["### Solve ( n(n + 1) = 10100 ): A Step-by-Step Guide to Double Both Sides", "Solving quadratic equations is a fundamental skill in algebra, and sometimes simplifying the equation first makes finding solutions easier. One common technique is multiplying both sides by a constant to eliminate fractions or simplify expressions. In this article, we’ll explore how multiplying both sides by 2 helps solve the equation:", "[\nn(n + 1) = 10100\n]", "---", "### Why Multiply Both Sides?", "Start by recognizing the left-hand side is a product:\n[\nn(n + 1) = 10100\n]\nWhile this equation is solvable, multiplying both sides by 2 simplifies the coefficients slightly and gives a cleaner quadratic form:\n[\n2n(n + 1) = 2 \ imes 10100\n]\n[\n\Rightarrow 2n(n + 1) = 20200\n]\nAlthough this doesn’t reduce the degree, it prepares the equation for expansion into a standard quadratic form:\n[\n2n^2 + 2n = 20200\n]\nThis makes it easier to rearrange into standard quadratic form ( ax^2 + bx + c = 0 ), allowing for direct factoring, completing the square, or applying the quadratic formula.", "---", "### Step-by-Step Solution", "1. Expand and Rearrange Equation\nStart from:\n[\nn(n + 1) = 10100\n]\nMultiply both sides by 2:\n[\n2n(n + 1) = 20200\n]\n[\n2n^2 + 2n - 20200 = 0\n]", "2. Simplify Quadratic Equation\nDivide every term by 2 to reduce coefficients:\n[\nn^2 + n - 10100 = 0\n]", "3. Factor or Apply Quadratic Formula\nWe now solve the quadratic equation:\n[\nn^2 + n - 10100 = 0\n]", "Since factoring is not straightforward here (no obvious integer pair multiplies to -10100 and adds to +1), use the quadratic formula:\n[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nWith ( a = 1 ), ( b = 1 ), ( c = -10100 ):\n[\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-10100)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 40400}}{2} = \frac{-1 \pm \sqrt{40401}}{2}\n]", "4. Compute Square Root\nFind ( \sqrt{40401} ). Testing nearby perfect squares or using a calculator gives:\n[\n\sqrt{40401} = 201\n]", "5. Solve for ( n )\n[\nn = \frac{-1 + 201}{2} = \frac{200}{2} = 100\n]\n[\nn = \frac{-1 - 201}{2} = \frac{-202}{2} = -101 \quad (\ ext{discard, as context implies } n > 0)\n]", "---", "### Final Answer", "The solution to the equation ( n(n + 1) = 10100 ) is:\n[\n\boxed{n = 100}\n]", "Verification:\n[\n100 \ imes 101 = 10100 \quad \ ext{✓}\n]", "Multiplying both sides by 2 was a strategic step that simplified direct expansion and enabled clear simplification to a manageable quadratic equation.", "---", "### Why This Method Matters", "- Reduces complexity before solving.\n- Clears coefficients to avoid decimal or compound terms.\n- Prepares equation for standard solving techniques.\n- Enhances clarity when applying the quadratic formula.", "Whether learning algebra or tackling real-world problems, mastering such algebraic manipulations builds confidence and accuracy in equation solving.", "---", "Keywords: Multiply both sides by 2, solve ( n(n+1) = 10100 ), quadratic equation, algebra tips, solve quadratic, factoring or quadratic formula, simplify equations."]
