Total area: \( (20 + 2x)(15 + 2x) = 504 \)

["Total Area Equation: Solving ( (20 + 2x)(15 + 2x) = 504 ) – A Step-by-Step Guide", "When it comes to geometry problems involving area, quadratic equations often arise—especially in real-world applications like architecture, interior design, and land planning. One common equation students encounter is:", "[\n(20 + 2x)(15 + 2x) = 504\n]", "This equation models a total area problem, where ( x ) represents an unknown dimension (like an extension or width), and parentheses define side lengths of a rectangular space. Understanding how to solve and interpret this equation unlocks insights into algebraic modeling and practical problem-solving.", "---", "### Understanding the Problem", "The expression ( (20 + 2x)(15 + 2x) ) expands to represent the area of a rectangle with one side fixed at 20 and the other pairing ( 2x ), while the adjacent side is ( 15 + 2x )—note both sides grow linearly with ( x ). The product equals 504 square units, reflecting a target area that could represent a building footprint, garden extension, or custom space.", "---", "### Step 1: Expand the Equation", "Start by expanding the left-hand side using the distributive property (FOIL method):", "[\n(20 + 2x)(15 + 2x) = 20 \cdot 15 + 20 \cdot 2x + 2x \cdot 15 + 2x \cdot 2x\n]", "[\n= 300 + 40x + 30x + 4x^2\n]", "[\n= 4x^2 + 70x + 300\n]", "Now set equal to 504:", "[\n4x^2 + 70x + 300 = 504\n]", "---", "### Step 2: Bring Equation to Standard Quadratic Form", "Subtract 504 from both sides:", "[\n4x^2 + 70x + 300 - 504 = 0\n]", "[\n4x^2 + 70x - 204 = 0\n]", "To simplify, divide every term by 2:", "[\n2x^2 + 35x - 102 = 0\n]", "This simplified form is easier to factor or apply the quadratic formula.", "---", "### Step 3: Solve the Quadratic Equation", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "where ( a = 2 ), ( b = 35 ), and ( c = -102 ).", "Calculate the discriminant:", "[\n\Delta = 35^2 - 4 \cdot 2 \cdot (-102) = 1225 + 816 = 2041\n]", "Then,", "[\nx = \frac{-35 \pm \sqrt{2041}}{4}\n]", "Since ( \sqrt{2041} \approx 45.18 ), compute both solutions:", "[\nx_1 = \frac{-35 + 45.18}{4} \approx \frac{10.18}{4} \approx 2.545\n]", "[\nx_2 = \frac{-35 - 45.18}{4} \approx \frac{-80.18}{4} \approx -20.045\n]", "---", "### Step 4: Select Valid Solution", "Area and dimensions must be positive. Since ( x \approx -20.045 ) gives negative lengths, discard it.", "Use ( x \approx 2.545 ). Both lengths become:", "- ( 20 + 2x \approx 20 + 5.09 = 25.09 )\n- ( 15 + 2x \approx 15 + 5.09 = 20.09 )", "Product:\n( 25.09 \ imes 20.09 \approx 504 ) ✓", "---", "### Why This Equation Matters", "Solving ( (20 + 2x)(15 + 2x) = 504 ) teaches key algebraic skills:", "- Polynomial expansion and simplification\n- Moving between equivalent forms (expanded and factored)\n- Applying the quadratic formula effectively\n- Interpreting mathematical models in real-world contexts", "This type of equation is frequently used in solving for unknown dimensions in business, engineering, and construction—making fluency here valuable for students and professionals alike.", "---", "### Final Thoughts", "Understanding and solving equations like ( (20 + 2x)(15 + 2x) = 504 ) strengthens algebraic reasoning and problem-solving acumen. Whether you're designing a room, planning a plot, or tackling SAT/ACT math, mastering these techniques empowers you to approach complex, real-world challenges with confidence.", "Keywords: total area equation, solving quadratic, algebra problem, (20 + 2x)(15 + 2x) = 504, step-by-step solution, quadratic formula, rectangular area, geometry applications, algebra help.", "---", "If you’re tackling this equation for school or work, remember: expanding carefully, checking solutions, and interpreting results in context are crucial steps to success."]