A rectangular garden measures 12 meters by 9 meters. If a path of uniform width is built around the garden, increasing the total area to 180 square meters, what is the width of the path?

["Title: How to Calculate the Width of a Uniform Garden Path Surrounding a 12m x 9m Garden", "When designing gardens, incorporating walkways or paths enhances both aesthetics and functionality. In this scenario, we explore a rectangular garden measuring 12 meters by 9 meters, with a uniformly wide path built around it, increasing the total area to 180 square meters. This article explains how to calculate the precise width of the path—an essential step for effective garden planning.", "### The Problem: Rectangular Garden with a Uniform Path", "The garden has dimensions 12 meters (length) by 9 meters (width). A path of equal width, let’s call it x meters, surrounds the entire garden. This increases each dimension:", "- New length: (12 + 2x) (path adds x meters on both sides)\n- New width: (9 + 2x)", "The total area, including the path, is given as 180 square meters.", "### Step-by-Step Calculation", "1. Write the formula for the total area:\n [\n \ ext{Total Area} = (12 + 2x)(9 + 2x) = 180\n ]", "2. Expand the expression:\n [\n (12 + 2x)(9 + 2x) = 12 \cdot 9 + 12 \cdot 2x + 2x \cdot 9 + 2x \cdot 2x\n ]\n [\n = 108 + 24x + 18x + 4x^2 = 4x^2 + 42x + 108\n ]", "3. Set up the equation:\n [\n 4x^2 + 42x + 108 = 180\n ]", "4. Simplify the equation:\n [\n 4x^2 + 42x + 108 - 180 = 0\n ]\n [\n 4x^2 + 42x - 72 = 0\n ]", "5. Divide the entire equation by 2 to simplify:\n [\n 2x^2 + 21x - 36 = 0\n ]", "6. Solve the quadratic equation using the quadratic formula:\n The quadratic formula is:\n [\n x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n ]\n Here, (a = 2), (b = 21), (c = -36):\n [\n x = \frac{-21 \pm \sqrt{21^2 - 4(2)(-36)}}{2 \cdot 2}\n ]\n [\n x = \frac{-21 \pm \sqrt{441 + 288}}{4}\n ]\n [\n x = \frac{-21 \pm \sqrt{729}}{4}\n ]\n [\n x = \frac{-21 \pm 27}{4}\n ]", "7. Calculate possible values:\n - (x = \frac{-21 + 27}{4} = \frac{6}{4} = 1.5)\n - (x = \frac{-21 - 27}{4} = \frac{-48}{4} = -12) (not physically meaningful, discard)", "Only (x = 1.5) meters is valid.", "### Final Answer", "The width of the uniform path around the 12m × 9m rectangular garden is 1.5 meters. This ensures the total area becomes 180 square meters, optimizing both space and usability.", "---", "This precise calculation helps homeowners, landscapers, and gardeners efficiently plan uniform pathways that complement garden layouts. Whether for accessibility, drainage, or aesthetic appeal, understanding path dimensions ensures a functional and beautiful outdoor space.", "Keywords: rectangular garden path, path width calculation, garden area expansion, uniform path around garden, 12m by 9m garden, landscape design."]