Wait: \( (20+2x)(15+2x) = 504 \)

["### Solving ( (20 + 2x)(15 + 2x) = 504 ): Step-by-Step Guide", "If you're tackling the equation ( (20 + 2x)(15 + 2x) = 504 ), you're likely navigating a common algebraic challenge involving quadratics. This equation provides a great opportunity to practice expanding expressions, setting up quadratic forms, and solving for ( x ).", "---", "#### Step 1: Expand the Left-Hand Side", "Start by expanding ( (20 + 2x)(15 + 2x) ) 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]", "Combine like terms:", "[\n= 300 + 70x + 4x^2\n]", "So the equation becomes:", "[\n4x^2 + 70x + 300 = 504\n]", "---", "#### Step 2: Simplify into Standard Quadratic Form", "Subtract 504 from both sides:", "[\n4x^2 + 70x + 300 - 504 = 0\n]", "[\n4x^2 + 70x - 204 = 0\n]", "You can simplify this further by dividing every term by 2:", "[\n2x^2 + 35x - 102 = 0\n]", "Now you have the standard quadratic equation:", "[\n2x^2 + 35x - 102 = 0\n]", "---", "#### Step 3: Solve the Quadratic Equation", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For ( a = 2 ), ( b = 35 ), ( c = -102 ):", "[\nx = \frac{-35 \pm \sqrt{35^2 - 4 \cdot 2 \cdot (-102)}}{2 \cdot 2}\n]", "[\nx = \frac{-35 \pm \sqrt{1225 + 816}}{4}\n]", "[\nx = \frac{-35 \pm \sqrt{2041}}{4}\n]", "Since ( \sqrt{2041} ) is not a perfect square, approximate for numerical values:", "[\n\sqrt{2041} \approx 45.18\n]", "So,", "[\nx_1 = \frac{-35 + 45.18}{4} \approx \frac{10.18}{4} \approx 2.545\n]\n[\nx_2 = \frac{-35 - 45.18}{4} \approx \frac{-80.18}{4} \approx -20.045\n]", "---", "#### Step 4: Verify the Solutions in the Original Equation", "Plug ( x \approx 2.545 ):", "[\n(20 + 2 \cdot 2.545)(15 + 2 \cdot 2.545) \approx (20 + 5.09)(15 + 5.09) = 25.09 \cdot 20.09 \approx 504\n]", "The solution checks.", "---", "#### Key Takeaways", "- Expanding binomials carefully avoids errors in computation.\n- Always simplify equations before applying formulas.\n- Quadratic equations often have irrational roots; use approximations if precision isn’t required.\n- Verification ensures accuracy in solving algebraic equations.", "---", "#### Why Solve Equations Like This?", "Mastering equations involving expanded binomials is crucial in algebra, geometry, and applied math. They appear in modeling real-world scenarios, from financial projections to physics problems.", "---", "Keywords: solve ((20 + 2x)(15 + 2x) = 504), quadratic equations, expand binomials, algebra practice, quadratic formula, solve (2x^2 + 35x - 102 = 0), algebraic solutions, mathematica, calculus prep.", "---", "If you found this guide helpful, share it to help others master quadratic equations!"]