The total area equation is \( (30 + 2x)(20 + 2x) = 936 \).

["# Solving the Total Area Equation: ( (30 + 2x)(20 + 2x) = 936 )", "Understanding how to solve equations involving areas is essential in algebra and real-world applications such as architecture, interior design, and land development. One classic example is solving for an unknown dimension when the total area of combined rectangular regions is known. In this article, we explore the total area equation ( (30 + 2x)(20 + 2x) = 936 ), how to simplify and solve it, and what the solution means.", "---", "## Understanding the Equation", "The equation\n[\n(30 + 2x)(20 + 2x) = 936\n]\nrepresents the total area of two rectangular regions:", "- One rectangle with length (30 + 2x) and width 1 (assumed fixed), giving area (30 + 2x).\n- Another rectangle with length (20 + 2x) and width 1, giving area (20 + 2x).", "The product of these two lengths equals 936 square units. Solving for (x) allows us to determine the unknown scaling factor in a proportional dimension.", "---", "## Step 1: Expand the Equation", "First, expand the left-hand side using the distributive property (FOIL method):", "[\n(30 + 2x)(20 + 2x) = 30 \cdot 20 + 30 \cdot 2x + 2x \cdot 20 + 2x \cdot 2x = 600 + 60x + 40x + 4x^2\n]", "Combine like terms:", "[\n4x^2 + 100x + 600 = 936\n]", "---", "## Step 2: Form a Quadratic Equation", "Subtract 936 from both sides to set the equation to zero:", "[\n4x^2 + 100x + 600 - 936 = 0\n]", "[\n4x^2 + 100x - 336 = 0\n]", "Divide every term by 4 to simplify:", "[\nx^2 + 25x - 84 = 0\n]", "---", "## Step 3: Solve the Quadratic Equation", "We solve ( x^2 + 25x - 84 = 0 ) using factoring, the quadratic formula, or completing the square. Here, we factor:", "Find two numbers that multiply to (-84) and add to (25). These are (28) and (-3):", "[\n(x + 28)(x - 3) = 0\n]", "Set each factor equal to zero:", "[\nx + 28 = 0 \quad \Rightarrow \quad x = -28\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "---", "## Step 4: Interpret the Solution", "The length (x = -28) is not physically meaningful in most real-world contexts since dimensions cannot be negative. Therefore, the valid solution is:", "[\nx = 3\n]", "---", "## Practical Implication", "Plugging (x = 3) back into the original dimensions:", "- First rectangle: (30 + 2(3) = 36)\n- Second rectangle: (20 + 2(3) = 26)", "Area = (36 \ imes 26 = 936), which confirms the solution.", "This demonstrates how scaling side lengths by the same factor (x) maintains proportionality while achieving a target total area.", "---", "## Conclusion", "Solving equations like ( (30 + 2x)(20 + 2x) = 936 ) illustrates key algebraic techniques—expansion, simplification, and quadratic solving—while applying directly to real-life area calculations. Always interpret solutions within context: reject negative or unrealistic answers. Mastering such problems strengthens problem-solving skills crucial for STEM fields and daily life applications involving geometry.", "---", "### SEO Keywords:\ntotal area equation, solve quadratic equation, algebra problems, area calculation, project dimensions, perpendicular bisector application, proportional geometry, algebra tutoring, math problem solved, quadratic forms, real-world application math", "---", "## Related Reading:\n- How to expand binomials and solve area problems\n- Applying quadratic equations to interior design and architecture\n- Step-by-step guide to solving word problems with algebra", "---", "By mastering equations such as ( (30 + 2x)(20 + 2x) = 936 ), you build a strong foundation for advanced mathematics and practical problem-solving skills."]