A rectangular garden measures 20 meters by 15 meters. If a path of uniform width is built around the garden, increasing the total area to 396 square meters, what is the width of the path?

["Title: How to Find the Uniform Width of a Path Surrounding a Rectangular Garden | Garden Area Expansion Calculations", "Creating a beautiful rectangular garden is a rewarding project, but often not enough space is reserved for accessible pathways — especially when planning a walking path around the garden. In this article, we’ll explore how to determine the exact width of a uniformly wide path surrounding a 20m by 15m garden, increasing the total area to 396 square meters. Whether you’re a gardener, landscaper, or DIY enthusiast, understanding the math behind garden path design ensures both functionality and beauty.", "### Garden Dimensions and Original Area", "Our garden is a perfect rectangle measuring 20 meters in length and 15 meters in width. Calculating its original area is simple:", "[\n\ ext{Original Area} = 20,m \ imes 15,m = 300,m²\n]", "### Expanded Area with Path", "Surrounding the garden, a path of uniform width ( x ) meters is added on all sides. This means the path adds ( 2x ) meters to both the length and the width:", "- New length = ( 20 + 2x ) meters\n- New width = ( 15 + 2x ) meters", "The total area of the garden plus path is given as 396 square meters, so:", "[\n(20 + 2x)(15 + 2x) = 396\n]", "### Expanding the Equation", "Expand the left-hand side using algebra:", "[\n(20 + 2x)(15 + 2x) = 20 \ imes 15 + 20 \ imes 2x + 2x \ imes 15 + 2x \ imes 2x = 300 + 40x + 30x + 4x^2\n]", "[\n4x^2 + 70x + 300 = 396\n]", "### Solving the Quadratic Equation", "Subtract 396 from both sides:", "[\n4x^2 + 70x + 300 - 396 = 0\n]", "[\n4x^2 + 70x - 96 = 0\n]", "Divide the entire equation by 2 to simplify:", "[\n2x^2 + 35x - 48 = 0\n]", "Now apply the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Where ( a = 2 ), ( b = 35 ), ( c = -48 ):", "[\nx = \frac{-35 \pm \sqrt{35^2 - 4(2)(-48)}}{2(2)} = \frac{-35 \pm \sqrt{1225 + 384}}{4} = \frac{-35 \pm \sqrt{1609}}{4}\n]", "Now compute ( \sqrt{1609} ). Since ( 40^2 = 1600 ) and ( 40.1^2 \approx 1608.01 ), ( \sqrt{1609} \approx 40.11 ).", "[\nx \approx \frac{-35 + 40.11}{4} = \frac{5.11}{4} \approx 1.2775\n]", "We discard the negative root since width cannot be negative.", "Thus, the width of the path is approximately 1.28 meters, or more precisely:", "[\nx = \frac{-35 + \sqrt{1609}}{4} \approx 1.28, \ ext{m}\n]", "For practical landscaping, rounding to 1.3 meters is reasonable — ensuring usability while harmonizing with the garden’s scale.", "### Final Thoughts", "Understanding how a uniform path expands garden area helps optimize outdoor spaces. Whether for walking, seating, or planting zones around the garden bed, precise calculations prevent wasted space and enhance flow. The rectangular garden’s expansion to 396 m² with a 1.28-meter-wide path demonstrates how design and math blend seamlessly for outdoor excellence.", "---", "Key Takeaways:", "- Expand garden dimensions by ( 2x ) on each side.\n- Total new area: ( (20+2x)(15+2x) = 396 ).\n- Solve the quadratic: ( 2x^2 + 35x - 48 = 0 ).\n- Path width ( x \approx 1.28 , \ ext{m} ).\n- Use algebraic methods to ensure accurate, scalable garden planning.", "Optimizing garden sizes and path dimensions fosters both aesthetics and practicality — a small math investment that yields big outdoor rewards!"]