Simplify: \(120v + 2400 + 180v = 5v^2 + 100v\).
["# Simplify (120v + 2400 + 180v = 5v^2 + 100v): A Step-by-Step Guide", "Simplifying complex equations is essential for solving problems in algebra, physics, and engineering. One common equation many encounter involves combining like terms and transforming expressions into standard quadratic form. In this article, we’ll fully simplify the equation:", "[\n120v + 2400 + 180v = 5v^2 + 100v\n]", "### Step 1: Combine Like Terms on the Left Side", "Start by identifying and combining the like terms on the left-hand side (LHS) of the equation. Focus on the terms involving (v):", "[\n120v + 180v = (120 + 180)v = 300v\n]", "So the LHS simplifies to:", "[\n2400 + 300v\n]", "Now rewrite the equation:", "[\n2400 + 300v = 5v^2 + 100v\n]", "### Step 2: Move All Terms to One Side", "To simplify further, move all terms to the left side by subtracting (5v^2 + 100v) from both sides:", "[\n2400 + 300v - 5v^2 - 100v = 0\n]", "Simplify the terms:", "- Combine (v) terms: (300v - 100v = 200v)\n- Keep (5v^2) as (-5v^2) (coefficient negative)", "Resulting equation:", "[\n-5v^2 + 200v + 2400 = 0\n]", "### Step 3: Simplify the Equation (Optional: Standard Form)", "To make the equation cleaner, multiply the entire expression by (-1) to eliminate the negative leading coefficient:", "[\n5v^2 - 200v - 2400 = 0\n]", "This is now in standard quadratic form:", "[\n5v^2 - 200v - 2400 = 0\n]", "You can divide every term by 5 to simplify further (this step not necessary but useful for easier solving):", "[\nv^2 - 40v - 480 = 0\n]", "### Final Simplified Equation\n[\n\boxed{5v^2 - 200v - 2400 = 0}\n]", "### Summary and Usage", "This quadratic equation results from simplifying a physical or applied math scenario involving voltage, power, and resistance or similar quantities. By simplifying:", "- Combine like terms on both sides\n- Move all terms to one side to form a standard quadratic equation\n- Simplify coefficients where possible", "This approach makes the equation ready for solving via factoring, completing the square, or the quadratic formula. Understanding this simplification process is key to tackling similar algebraic equations efficiently.", "---", "Keywords: simplify quadratic equation, solve $120v + 2400 + 180v = 5v^2 + 100v$, simplify $5v^2 - 200v - 2400 = 0$, algebraic simplification, quadratic equations."]
