New area = (20 + 2w)(15 + 2w) = 396.

["### Solving the Quadratic Equation: (20 + 2w)(15 + 2w) = 396 – A Step-by-Step Guide", "When faced with the quadratic expression (20 + 2w)(15 + 2w) = 396, solving for variable w becomes a clear algebraic challenge—one perfectly suited for step-by-step explanation. Whether you're a student tackling math homework or a professional exploring algebraic modeling, understanding how to expand, simplify, and solve such equations is essential.", "This article walks you through solving (20 + 2w)(15 + 2w) = 396, covering:", "- Expanding the expression\n- Simplifying the equation\n- Solving for w using algebraic techniques\n- Verifying the results", "Let’s dive in.", "---", "#### Step 1: Expand the Left-Hand Side", "Start by expanding the product:\n[\n(20 + 2w)(15 + 2w)\n]", "Apply the distributive property (FOIL method):\n- First: (20 \cdot 15 = 300)\n- Outer: (20 \cdot 2w = 40w)\n- Inner: (2w \cdot 15 = 30w)\n- Last: (2w \cdot 2w = 4w^2)", "Add them together:\n[\n300 + 40w + 30w + 4w^2 = 4w^2 + 70w + 300\n]", "So now the equation becomes:\n[\n4w^2 + 70w + 300 = 396\n]", "---", "#### Step 2: Bring All Terms to One Side", "Subtract 396 from both sides to form a standard quadratic equation:\n[\n4w^2 + 70w + 300 - 396 = 0\n\Rightarrow 4w^2 + 70w - 96 = 0\n]", "---", "#### Step 3: Simplify the Equation (Optional)", "Since all coefficients are divisible by 2, divide the entire equation by 2 to simplify:\n[\n2w^2 + 35w - 48 = 0\n]", "Now you’re working with a cleaner standard quadratic equation:\n[\n2w^2 + 35w - 48 = 0\n]", "---", "#### Step 4: Solve the Quadratic Using the Quadratic Formula", "Use the quadratic formula:\n[\nw = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nwhere (a = 2), (b = 35), and (c = -48).", "Calculate the discriminant:\n[\n\Delta = b^2 - 4ac = 35^2 - 4(2)(-48) = 1225 + 384 = 1609\n]", "Now compute the roots:\n[\nw = \frac{-35 \pm \sqrt{1609}}{4}\n]", "Note: (\sqrt{1609}) is an irrational number (~40.11), so the solutions are:\n[\nw = \frac{-35 + \sqrt{1609}}{4}, \quad w = \frac{-35 - \sqrt{1609}}{4}\n]", "---", "#### Step 5: Approximate the Solutions (Useful for Practical Use)", "To interpret the results in real-world scenarios (e.g., area modeling), approximate the roots:\n[\n\sqrt{1609} \approx 40.11\n\Rightarrow w \approx \frac{-35 + 40.11}{4} = \frac{5.11}{4} \approx 1.28\n]\nand\n[\nw \approx \frac{-35 - 40.11}{4} = \frac{-75.11}{4} \approx -18.78\n]", "Only the positive root (w \approx 1.28) is meaningful in most practical applications (e.g., dimensions, growth models), since negative values in such contexts often lack physical significance.", "---", "#### Step 6: Verify the Solution by Substitution", "Plug (w \approx 1.28) back into the original expression:\n[\n(20 + 2 \cdot 1.28)(15 + 2 \cdot 1.28) = (20 + 2.56)(15 + 2.56) = 22.56 \cdot 17.56 \approx 396\n]\nThis confirms the solution is accurate.", "---", "### Why Does This Equation Matter?", "Equations of the form (a + bw)(c + dw) = constant often model real-life situations — such as calculating area in expanded dimensions, profit projections with variable costs, or engineering design parameters. Understanding how to solve these deepens algebraic fluency and problem-solving skills.", "---", "#### Summary", "- Expanded: ((20 + 2w)(15 + 2w) = 4w^2 + 70w + 300)\n- Simplified: (2w^2 + 35w - 48 = 0)\n- Solved via quadratic formula:\n [\n w = \frac{-35 \pm \sqrt{1609}}{4}\n ]\n- Verified approximate solution yields (w \approx 1.28)\n- Ideal for modeling expanded area or joint growth rates", "---", "### SEO Keywords to Optimize This Article:\n- Solve quadratic equation (20 + 2w)(15 + 2w) = 396\n- Expand and simplify (20 + 2w)(15 + 2w)\n- Algebraic methods for quadratic equations\n- Solve quadratic formula step-by-step\n- Practical application of quadratic equations\n- How to solve (a + bw)(c + dw) = k", "---", "Whether you’re solving for precision in calculations, learning algebra basics, or modeling real-world systems, mastering this type of equation empowers your analytical toolkit. Start expanding, simplifying, and solving—your next math breakthrough is just one equation away!"]