Set up the equation for total area: \((15 + 2x)(10 + 2x) = 286\).
["Title: How to Set Up the Equation for Total Area: ( (15 + 2x)(10 + 2x) = 286 )", "Meta Description:\nLearn how to properly set up the area equation ((15 + 2x)(10 + 2x) = 286) step-by-step. This guide explains each term and why they matter for solving quadratic area problems.", "---", "When solving geometry problems involving area—especially when dimensions relate to a variable—setting up the correct equation is crucial. One common scenario involves compound shapes, such as a figure formed by attaching a rectangular extension to a base rectangle. In this case, we’re given:", "[\n(15 + 2x)(10 + 2x) = 286\n]", "This equation models a situation where one dimension of a base rectangle is (15 + 2x) (including an adjustable extension of length (x)), and another dimension is (10 + 2x), yielding a total area of 286 square units.", "---", "### Step-by-Step Setup of the Equation", "1. Understand the geometry\n Imagine a rectangle with length (15 + 2x) and width (10 + 2x). The factor of 2 arises because the extension adds (x) to both ends, effectively doubling the adjustment. Multiplying these gives the total area.", "2. Write the area expression\n The area (A) of a rectangle is length multiplied by width. Thus,", "[\n A = (15 + 2x)(10 + 2x)\n ]", "3. Substitute the known total area\n Since the total area is 286, substitute this value into the equation:", "[\n (15 + 2x)(10 + 2x) = 286\n ]", "4. Expand and simplify (optional for setup)\n Though not needed just to set up the equation, expanding yields:", "[\n 150 + 30x + 20x + 4x^2 = 286\n ]\n [\n 4x^2 + 50x + 150 = 286\n ]\n [\n 4x^2 + 50x - 136 = 0\n ]", "But the original factored form correctly captures the structural relationship.", "---", "### Why This Form Works", "The expression ((15 + 2x)(10 + 2x)) accounts for incremental growth: doubling (x) reflects symmetric expansion—common in models involving uniform additions. By expressing both length and width in terms of (x), the equation remains flexible and solvable using algebraic techniques like factoring or the quadratic formula.", "---", "### Next Steps: Solving the Equation", "Once correctly set up:", "[\n(15 + 2x)(10 + 2x) = 286\n]", "This leads to a quadratic equation that opens upward (positive (x^2) coefficient), allowing for real solutions if the area matches the expanded dimensions. Solve via factoring, completing the square, or using the quadratic formula to find (x), then substitute back to determine exact dimensions.", "---", "### Summary", "Setting up a total area equation involves:\n- Identifying the two variable-expressing dimensions,\n- Multiplying them to represent total area,\n- Substituting the known constant value.", "The equation ((15 + 2x)(10 + 2x) = 286) elegantly models this scenario. Mastering such setups strengthens problem-solving across algebra, physics, and engineering applications.", "Keywords: equation setup, total area, quadratic equation, (15 + 2x)(10 + 2x) = 286, algebra, geometry, problem-solving, area calculation.", "---", "If you want help solving this quadratic or finding (x), let me know—I’m happy to walk through the next steps!"]
