Multiply both sides by 2: n(n + 1) = 420.

["Multiply Both Sides by 2: n(n + 1) = 420 – Solving a Key Algebraic Equation", "When learning algebra, one of the most common tasks is simplifying and solving equations like n(n + 1) = 420. This type of equation appears often in math problems and real-world applications. One effective technique to solve such expressions is by multiplying both sides by a strategic value—in this case, 2—to simplify the structure and prepare the equation for factoring.", "---", "### Why Multiply Both Sides by 2?", "The original equation is:", "$$\nn(n + 1) = 420\n$$", "This form is a quadratic expression in factored form, but it’s not yet in standard quadratic form. Multiplying both sides by 2 helps eliminate fractions (if any) and simplifies manipulation—though here it mainly prepares the equation for easier factoring or completing the square later on.", "Multiplying by 2 gives:", "$$\n2n(n + 1) = 840\n$$", "Now expand:", "$$\n2n^2 + 2n = 840\n$$", "This makes the equation easier to handle if you choose to rewrite it as a standard quadratic:", "$$\n2n^2 + 2n - 840 = 0\n$$", "From here, you can divide the entire equation by 2 to simplify coefficients:", "$$\nn^2 + n - 420 = 0\n$$", "Now the equation is in standard quadratic form:\n$$\nn^2 + n - 420 = 0\n$$", "---", "### Step-by-Step Solution", "Next, factor the quadratic equation. We need two numbers that multiply to -420 and add to +1.", "After checking factor pairs, we find:", "$$\n(n + 21)(n - 20) = 0\n$$", "Set each factor equal to zero:", "- $ n + 21 = 0 \Rightarrow n = -21 $\n- $ n - 20 = 0 \Rightarrow n = 20 $", "---", "### Interpreting the Solutions", "Since n often represents a count (like number of items, people, or time), we discard non-positive solutions if context requires positive integers. Thus, the meaningful solution is:", "$$\nn = 20\n$$", "Verify by plugging back into the original equation:", "$$\n20(20 + 1) = 20 \ imes 21 = 420 \quad \ ext{✓ Correct!}\n$$", "---", "### Final Thoughts", "Multiplying both sides of n(n + 1) = 420 by 2 is a practical step that helps streamline into a familiar quadratic form. This method reinforces skills in algebraic manipulation, factoring, and equation solving—essential foundations in math education.", "Whether you're solving math problems for homework or understanding core algebraic principles, mastering this technique makes complex equations much more approachable. Keep practicing, and soon multiplying both sides will feel intuitive!", "---", "### Key Takeaway", "To solve equations like n(n + 1) = 420, multiply both sides by 2 to convert to standard quadratic form, simplify, and factor. This transforms abstract expressions into solvable equations—empowering confident problem-solving in algebra!", "---", "Keywords for SEO:\nMultiply both sides by 2, solve quadratic equation, n(n + 1) = 420, algebra tips, quadratic factoring, algebraic manipulation, how to solve equations, standard quadratic form, math problem solving."]