Expand: \( 2n^2 + 4n = 220 \).

["# Solve the Equation ( 2n^2 + 4n = 220 ): A Complete Step-by-Step Guide", "Solving quadratic equations is a fundamental algebra skill, and understanding how to solve equations like ( 2n^2 + 4n = 220 ) opens the door to tackling more complex mathematical problems. In this comprehensive guide, we’ll walk through finding the values of ( n ) that satisfy this equation using logical steps and clear explanations tailored for beginners and learners of all levels.", "## Understanding the Equation", "We begin with the equation:\n[\n2n^2 + 4n = 220\n]\nThis is a quadratic equation in standard form, though it’s easier to solve when simplified into a classification:", "### Step 1: Rewrite in Standard Quadratic Form", "To solve ( 2n^2 + 4n = 220 ), subtract 220 from both sides:\n[\n2n^2 + 4n - 220 = 0\n]", "Now the equation is in standard quadratic form:\n[\nan^2 + bn + c = 0\n]\nwhere\n- ( a = 2 )\n- ( b = 4 )\n- ( c = -220 )", "---", "## Step 2: Simplify the Equation", "Before solving, simplify the equation by dividing all terms by the greatest common divisor (GCD), which is 2:\n[\n\frac{2n^2 + 4n - 220}{2} = \frac{0}{2}\n]\n[\nn^2 + 2n - 110 = 0\n]", "Now we solve the simplified quadratic equation:\n[\nn^2 + 2n - 110 = 0\n]", "---", "## Step 3: Factor the Quadratic (If Possible)", "Look for two numbers that multiply to ( -110 ) and add to ( 2 ).\nAfter testing factor pairs of ( -110 ), we find ( 11 ) and ( -10 ):\n[\n11 \ imes (-10) = -110 \quad \ ext{and} \quad 11 + (-10) = 1 \quad \ ext{(Not correct!)}\n]\nActually, try ( 11 ) and ( -10 ) carefully — no, correction:\nCorrect factor pair: ( 11 ) and ( -10 ) sum to 1 — wrong\nCorrect pair: Try ( 11 ) and ( -10 ) → invalid; correct pair is ( 11 ) and ( -10 )? No.", "Actually, factoring by trial or the quadratic formula is more reliable here.", "---", "## Step 4: Use the Quadratic Formula (Most Reliable Method)", "Since factoring is not straightforward, apply the quadratic formula:\n[\nn = \frac{ -b \pm \sqrt{b^2 - 4ac} }{ 2a }\n]\nPlug in: ( a = 1 ), ( b = 2 ), ( c = -110 ):\n[\nn = \frac{ -2 \pm \sqrt{(2)^2 - 4(1)(-110)} }{ 2(1) }\n]\n[\nn = \frac{ -2 \pm \sqrt{4 + 440} }{ 2 }\n]\n[\nn = \frac{ -2 \pm \sqrt{444} }{ 2 }\n]", "Simplify ( \sqrt{444} ):\n[\n\sqrt{444} = \sqrt{4 \ imes 111} = 2\sqrt{111}\n]", "So:\n[\nn = \frac{ -2 \pm 2\sqrt{111} }{ 2 } = -1 \pm \sqrt{111}\n]", "---", "## Step 5: Final Solutions", "Thus, the exact solutions are:\n[\nn = -1 + \sqrt{111} \quad \ ext{and} \quad n = -1 - \sqrt{111}\n]", "Since ( \sqrt{111} \approx 10.54 ), approximate numerical solutions are:\n[\nn \approx -1 + 10.54 = 9.54 \quad \ ext{and} \quad n \approx -1 - 10.54 = -11.54\n]", "Only real solutions are ( n = -1 + \sqrt{111} ) and ( n = -1 - \sqrt{111} ), with ( n \approx 9.54 ) being the positive valid one depending on context.", "---", "## Why This Equation Matters", "Understanding how to solve equations like ( 2n^2 + 4n = 220 ):\n- Strengthens algebraic manipulation skills.\n- Provides a foundation for modeling real-world problems (e.g., projectile motion, profit calculations).\n- Encourages the use of both factoring and quadratic formula for robust problem-solving.", "---", "## Tips for Solving Quadratic Equations:", "- Always simplify before applying formulas.\n- Check whether factoring is easy or if the quadratic formula is quicker.\n- Use the discriminant ( D = b^2 - 4ac ) to determine solution type:\n - ( D > 0 ): two real solutions\n - ( D = 0 ): one real solution\n - ( D < 0 ): no real solutions\n- Estimate roots using irrational values if exact roots are messy.", "---", "## Further Reading & Related Topics", "- Solve linear and quadratic inequalities\n- Graph quadratic functions and analyze roots\n- Apply quadratic formulas in physics in problems involving motion\n- Explore completing the square as an alternative solving method", "---", "## Conclusion", "Solving ( 2n^2 + 4n = 220 ) involves simplifying the quadratic equation and applying the quadratic formula. While the exact solutions involve radicals, understanding the process equips you to handle similar problems confidently. Keep practicing algebra — each equation brings you closer to mastery!", "---", "Keywords for this article:\nquadratic equation solution, solve (2n^2 + 4n = 220), quadratic formula example, algebra guide, solving quadratic equations step-by-step, step-by-step quadratic solutions, expand and solve, use quadratic form, real solutions of quadratic, algebraic methods.", "---", "If you want help checking steps or applying this to real-world scenarios, feel free to search — mastering quadratics unlocks powerful problem-solving skills!"]