Ãquation du volume : \( 2w imes w imes (w - 5) = 1200 \Rightarrow 2w^2(w - 5) = 1200 \).
["Understanding the Equation: Volume, Area, and Algebra – Solving (2w(w - 5) = 1200)", "When solving real-world problems involving geometry and algebra, equations like (2w(w - 5) = 1200) frequently appear. This equation models scenarios such as optimizing the volume of a three-dimensional shape, where (w) represents a width or a dimension of an object. In this SEO-optimized article, we’ll walk you through solving the equation, interpreting its meaning, and applying it in practical contexts.", "---", "### What Does (2w(w - 5) = 1200) Represent?", "The equation (2w(w - 5) = 1200) might model a situation where volume or area depends nonlinearly on a width (w). Let’s unpack this.", "- (w): width (in meters, centimeters, etc. – units depend on context)\n- (w - 5): a related dimension, possibly a reduced length or adjustment factor\n- (2w(w - 5)): a product reflecting a multi-dimensional area volume (for example, a quadrilateral prism or a polynomial-based model)\n- Equals 1200: a target value, such as volume or area, being analyzed", "Such expressions arise in engineering, architecture, and optimization problems where geometry and algebra combine.", "---", "### Step-by-Step Solution of the Equation", "We begin with:\n[\n2w(w - 5) = 1200\n]", "Step 1: Expand the left-hand side\nDistribute (2w) across (w - 5):\n[\n2w^2 - 10w = 1200\n]", "Step 2: Bring all terms to one side\n[\n2w^2 - 10w - 1200 = 0\n]", "Step 3: Simplify the quadratic\nDivide every term by 2 to reduce complexity:\n[\nw^2 - 5w - 600 = 0\n]", "Step 4: Solve using the quadratic formula\nFor (aw^2 + bw + c = 0), the solutions are:\n[\nw = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nHere, (a = 1), (b = -5), (c = -600):\n[\nw = \frac{5 \pm \sqrt{(-5)^2 - 4(1)(-600)}}{2(1)} = \frac{5 \pm \sqrt{25 + 2400}}{2} = \frac{5 \pm \sqrt{2425}}{2}\n]", "Simplify (\sqrt{2425}): note (2425 = 25 \ imes 97), so:\n[\n\sqrt{2425} = 5\sqrt{97}\n]\nThus,\n[\nw = \frac{5 \pm 5\sqrt{97}}{2} = \frac{5(1 \pm \sqrt{97})}{2}\n]", "Since (w) represents a physical dimension, only the positive root is meaningful:\n[\nw = \frac{5(1 + \sqrt{97})}{2}\n]", "Numerically, (\sqrt{97} \approx 9.849), so:\n[\nw \approx \frac{5(1 + 9.849)}{2} = \frac{5 \ imes 10.849}{2} \approx \frac{54.245}{2} \approx 27.12\n]", "---", "### Interpretation and Real-World Application", "The solution (w \approx 27.12) (cm, meters, etc.) means: when the width of a physical object is approximately 27.12 units, multiplying (2w(w - 5)) yields 1200.", "Example Application:\nImagine a rectangular prism where one dimension is width (w), and volume/area depends on (w(w - 5)) due to design constraints or material efficiency. At (w \approx 27.12), optimizing for a target performance or capacity (reflected as 1200) becomes feasible.", "---", "### Why This Equation Matters in Geometry and Algebra", "- Quadratic Growth Models: The product (w(w - 5)) reflects quadratic dependencies, common in scale-up problems.\n- Parametric Modeling: Such equations help model adjustable dimensions in construction and manufacturing.\n- Algebraic Fluency: Mastering solving these instills skills for tackling complex real-world optimization and design challenges.", "---", "### Tips for Solvers and Learners", "1. Verify Units: Ensure all terms use consistent units—critical in applied problems.\n2. Check Positive Solutions: Only positive roots apply to physical dimensions.\n3. Use Software Wisely: Tools like WolframAlpha or graphing calculators speed up verification.\n4. Explore Graphing: Plotting (y = 2w(w - 5)) vs (w) reveals root locations and behavior.", "---", "### Conclusion", "Solving the equation (2w(w - 5) = 1200) goes beyond algebra—it connects abstract math to practical engineering and design. From calculating optimal dimensions to modeling real systems, understanding such equations empowers learners to translate theory into tangible solutions.", "Keywords: Equation (2w(w - 5) = 1200), solve quadratic, real-world applications, algebra problem, volume modeling, quadratic roots, geometry algebra, mathematical modeling.", "---", "Whether you’re a student, teacher, or professional, mastering equations like (2w(w - 5) = 1200) builds a strong foundation for tackling complex STEM challenges. Keep practicing, stay curious, and apply this algebraic insight wherever geometry and numbers meet."]
