Expanding: 96 + 24x + 16x + 4x^2 = 200
["Expanding & Solving the Equation: 96 + 24x + 16x + 4x² = 200", "In today’s math-focused world, solving equations efficiently is a valuable skill—whether you're a student, educator, or professional tackling real-world problems. One interesting algebraic challenge combines expanding expression simplification with equation solving:", "96 + 24x + 16x + 4x² = 200", "This article walks you through expanding the expression (when needed), simplifying, and solving for x—all while highlighting best practices in algebraic manipulation and equation solving.", "---", "### Understanding the Equation: Expansion and Simplification", "At first glance, the equation is:\n96 + 24x + 16x + 4x² = 200", "While this equation doesn’t require expansion in its current form—since most terms are already combined except the 4x² term—it’s a great opportunity to revisit foundational algebra steps: combining like terms and rearranging to standard quadratic form.", "Start by combining like terms:\n24x + 16x = 40x", "So the equation becomes:\n4x² + 40x + 96 = 200", "Now, simplify by subtracting 200 from both sides:\n4x² + 40x + 96 - 200 = 0\n4x² + 40x - 104 = 0", "This is now the standard quadratic form:\nax² + bx + c = 0\nwhere a = 4, b = 40, c = -104", "---", "### Solving the Quadratic Equation", "To solve 4x² + 40x - 104 = 0, we can use the quadratic formula:\nx = [-b ± √(b² - 4ac)] / (2a)", "Plug in the values:\n– b = –40\n– a = 4\n– c = –104", "First, calculate the discriminant:\nΔ = b² - 4ac = (40)² - 4(4)(-104) = 1600 + 1664 = 3264", "Now compute:\nx = [ -40 ± √3264 ] / (8)", "Simplify √3264. Try factoring:\n3264 = 16 × 204 = 16 × 4 × 51 = so √3264 = 4 × 2 × √51 = 8√51\n(Since 51 = 3 × 17, no further simplification possible.)", "Thus:\nx = [ -40 ± 8√51 ] / 8 = (-5 ± √51) / 2", "So the two solutions are:\nx₁ = $\frac{-5 + \sqrt{51}}{2}$\nx₂ = $\frac{-5 - \sqrt{51}}{2}$", "---", "### Practical Application and Key Takeaways", "This problem demonstrates how:\n- Combining like terms simplifies expressions before solving.\n- Rearranging terms uses standard quadratic form for consistent solving.\n- The quadratic formula is essential when factoring is difficult.", "Whether applying math in physics modeling, economics, or engineering, mastering such steps enables clearer problem-solving and accurate results.", "---", "### Final Notes", "- Always verify solutions by plugging them back into the original equation.\n- Use tools like graphing calculators or algebra software (e.g., Wolfram Alpha) to confirm answers visually.\n- Understanding expression expansion and simplification is foundational long before coding or data analysis.", "Stay sharp with your algebra—every equation solved opens the door to deeper mathematical insight!", "---", "Keywords: solving quadratic equations, expanding algebraic expressions, simplifying equations, quadratic formula, 4x² + 40x - 104 = 0, algebra tutorial, workflow for solving equations, expanding and solving quadratic, math problem solving, mathematics education, solving for x algebraically", "---", "Related Reads:\n- How to Solve Quadratic Equations by Factoring\n- Step-by-Step Quadratic Formula Guide\n- Applications of Algebra in Real-World Science", "---", "By learning to expand, simplify, and solve systematically, you build a strong foundation for tackling complex mathematical models ahead. Keep practicing—and expanding your math toolkit!"]
