Innere Abmessungen: \( (20 - 2x) \times (15 - 2x) = 240 \).

["Title: Solving Innere Abmessungen: How to Find the Correct Interior Dimensions with the Equation ( (20 - 2x)(15 - 2x) = 240 )", "---", "### Introduction", "When working with interior design, architecture, or construction projects, finding the correct inner dimensions of a space is crucial. One common mathematical model used to determine these dimensions is the quadratic equation:", "[\n(20 - 2x)(15 - 2x) = 240\n]", "This equation helps determine the optimal “x” value that adjusts wall projections inward—typically minimizing wasted space while meeting design or structural requirements. In this article, we explore how to solve for (x), interpret the inner dimensions, and apply the solution practically in real-world scenarios.", "---", "### Understanding the Equation", "The expression ( (20 - 2x)(15 - 2x) ) represents the product of two inward-adjusted side lengths, where (x) defines how much the side projections reduce from the original measurements. The equation sets this product equal to the target area (240), balancing interior space efficiency.", "Expanding the expression:", "[\n(20 - 2x)(15 - 2x) = 300 - 40x - 30x + 4x^2 = 4x^2 - 70x + 300\n]", "Set equal to 240:", "[\n4x^2 - 70x + 300 = 240\n]", "Simplify:", "[\n4x^2 - 70x + 60 = 0\n]", "---", "### Step-by-Step Solution", "Step 1: Simplify the quadratic equation", "[\n4x^2 - 70x + 60 = 0\n]", "Divide through by 2 to make it simpler:", "[\n2x^2 - 35x + 30 = 0\n]", "Step 2: Use the quadratic formula", "The quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, (a = 2), (b = -35), (c = 30). Plug in the values:", "[\nx = \frac{35 \pm \sqrt{(-35)^2 - 4 \cdot 2 \cdot 30}}{2 \cdot 2}\n]\n[\nx = \frac{35 \pm \sqrt{1225 - 240}}{4}\n]\n[\nx = \frac{35 \pm \sqrt{985}}{4}\n]", "Calculate (\sqrt{985} \approx 31.38):", "[\nx = \frac{35 \pm 31.38}{4}\n]", "Compute both roots:", "- (x_1 = \frac{35 + 31.38}{4} = \frac{66.38}{4} \approx 16.60)\n- (x_2 = \frac{35 - 31.38}{4} = \frac{3.62}{4} \approx 0.905)", "Step 3: Choose the physically meaningful root", "Since the original dimensions are 20 and 15, reducing side projections beyond those values would yield negative or zero measurements. Thus, (x_2 \approx 0.905) is the valid solution.", "---", "### Calculating the Inner Dimensions", "Use (x \approx 0.905) to compute:", "[\n\ ext{Width} = 20 - 2x = 20 - 2(0.905) = 20 - 1.81 = 18.19 , \ ext{units}\n]\n[\n\ ext{Length} = 15 - 2x = 15 - 1.81 = 13.19 , \ ext{units}\n]", "Verify area:\n(18.19 \ imes 13.19 \approx 240) — correct.", "---", "### Practical Applications", "- Architectural Planning: Ensures interior layouts use exact square footage, minimizing wasted space.\n- Furniture Layout: Helps decide optimal placements without overcrowding.\n- Energy Efficiency: Accurate inner dimensions improve HVAC sizing and insulation calculations.\n- Construction Estimations: Enables precise material order based on real usable area.", "---", "### Conclusion", "Solving for inner dimensions using the equation ( (20 - 2x)(15 - 2x) = 240 ) involves transforming a word problem into a quadratic equation, solving algebraically, and validating physical relevance. The smaller valid root (x \approx 0.905) tailors the interior 'inward' to achieve a target area efficiently. Mastering such equations empowers designers and builders to handle complex space optimization with confidence.", "---", "Keywords: innere abmessungen, ( (20 - 2x) \ imes (15 - 2x) = 240 ), quadratic equation, interior dimensions, space optimization, algebra in architecture, interior design mathematics", "Meta Description: Learn how to solve ( (20 - 2x)(15 - 2x) = 240 ) for inner room dimensions, a key technique in architectural planning and space optimization. Find step-by-step solution and real-world applications."]