\( (20 + 2x)(15 + 2x) = 504 \)

["# Solving ( (20 + 2x)(15 + 2x) = 504 ): A Step-by-Step Algebra Guide", "When faced with equations like ( (20 + 2x)(15 + 2x) = 504 ), solving for ( x ) may seem challenging at first. However, with a structured approach and a few key algebraic steps, this quadratic equation becomes manageable and even enjoyable. This article walks you through solving the equation ( (20 + 2x)(15 + 2x) = 504 ), explaining the importance of proper expansion, simplifying, and factoring — essential skills for mastering quadratic equations.", "---", "## Why This Equation Matters", "Quadratic equations appear frequently in algebra, physics, and real-world applications such as area computations and motion problems. Solving equations of the form ( (a + 2x)(b + 2x) = C ) helps strengthen core problem-solving skills. By working through this problem, you improve your ability to expand expressions, create quadratics systematically, and factor or use the quadratic formula effectively.", "---", "## Step 1: Expand the Left Side", "Start by multiplying the binomials on the left-hand side:", "[\n(20 + 2x)(15 + 2x)\n]", "Apply the distributive property (also known as FOIL):", "- First: ( 20 \ imes 15 = 300 )\n- Outer: ( 20 \ imes 2x = 40x )\n- Inner: ( 2x \ imes 15 = 30x )\n- Last: ( 2x \ imes 2x = 4x^2 )", "Now combine all terms:", "[\n4x^2 + 40x + 30x + 300 = 4x^2 + 70x + 300\n]", "So the equation becomes:", "[\n4x^2 + 70x + 300 = 504\n]", "---", "## Step 2: Bring All Terms to One Side", "Subtract 504 from both sides to form a standard quadratic equation:", "[\n4x^2 + 70x + 300 - 504 = 0\n]", "[\n4x^2 + 70x - 204 = 0\n]", "To simplify, divide every term by 2:", "[\n2x^2 + 35x - 102 = 0\n]", "Now you have a simplified quadratic to solve.", "---", "## Step 3: Solve the Quadratic Equation", "Use one of the standard methods: factoring, completing the square, or the quadratic formula. In this case, factoring may not be straightforward, so we apply the quadratic formula:", "For any quadratic equation ( ax^2 + bx + c = 0 ), the solutions are:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 2 ), ( b = 35 ), ( c = -102 ).", "Calculate the discriminant:", "[\nb^2 - 4ac = 35^2 - 4(2)(-102) = 1225 + 816 = 2041\n]", "Note: ( \sqrt{2041} ) is not a perfect square, so the solutions will be irrational.", "Now compute:", "[\nx = \frac{-35 \pm \sqrt{2041}}{4}\n]", "Approximate ( \sqrt{2041} \approx 45.18 ) (using calculator):", "[\nx \approx \frac{-35 \pm 45.18}{4}\n]", "Two approximate solutions:", "[\nx_1 \approx \frac{-35 + 45.18}{4} = \frac{10.18}{4} \approx 2.545\n]\n[\nx_2 \approx \frac{-35 - 45.18}{4} = \frac{-80.18}{4} \approx -20.045\n]", "---", "## Step 4: Final Answer and Verification", "The exact solutions are:", "[\nx = \frac{-35 \pm \sqrt{2041}}{4}\n]", "For approximations:", "- ( x \approx 2.55 )\n- ( x \approx -20.05 )", "To verify, substitute back into the original equation ( (20 + 2x)(15 + 2x) ):", "- For ( x \approx 2.55 ):\n ( 20 + 2(2.55) = 25.1 )\n ( 15 + 2(2.55) = 20.1 )\n ( 25.1 \ imes 20.1 \approx 505.01 ) — close to 504 (rounding error)", "Thus, the solutions are accurate.", "---", "## Why This Matters Beyond the Equation", "Mastering how to expand, simplify, and solve quadratic expressions like this builds confidence in handling more complex functions and real-life modeling problems. Whether calculating areas, analyzing motion, or designing structures, these algebraic techniques form the foundation of higher mathematics.", "---", "## Conclusion", "Equation: ( (20 + 2x)(15 + 2x) = 504 )\nExpanded form: ( 4x^2 + 70x + 300 = 504 )\nSimplified: ( 2x^2 + 35x - 102 = 0 )\nSolution: ( x = \frac{-35 \pm \sqrt{2041}}{4} )", "By following a logical sequence — expand, simplify, apply quadratic formula — you’ve transformed a challenging expression into clear, solvable steps. Keep practicing this method, and quadratic equations will go from intimidating to manageable.", "---", "### Related Topics:\n- How to expand binomials\n- Quadratic formula explained\n- Solving real-world problems with algebra\n- Improving algebra skills step-by-step", "---", "Keywords: quadratic equation ( (20 + 2x)(15 + 2x) = 504 ), solve ( 4x^2 + 70x + 300 = 504 ), factoring quadratics, solving ( 2x^2 + 35x - 102 = 0 ), algebraic techniques, step-by-step algebra guide, quadratic solutions with irrational roots.", "---", "Meta description:\nLearn how to solve ( (20 + 2x)(15 + 2x) = 504 ) by expanding, simplifying, and using the quadratic formula. Step-by-step algebra with exact and approximate solutions."]