4x^2 - 78x + 108 = 0

["# Solving the Quadratic Equation: 4x² – 78x + 108 = 0", "Understanding how to solve quadratic equations is fundamental in algebra, and equations of the form ax² + bx + c = 0 lie at the heart of many mathematical applications. In this article, we will explore how to solve the quadratic equation 4x² – 78x + 108 = 0 step-by-step, analyze its roots, and provide insights into its real-world relevance.", "---", "## Step-by-Step Solution of 4x² – 78x + 108 = 0", "### 1. Identify the coefficients\nThe general form of a quadratic equation is:", "ax² + bx + c = 0", "For the equation 4x² – 78x + 108 = 0,\n- a = 4\n- b = –78\n- c = 108", "---", "### 2. Apply the Quadratic Formula\nThe quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute a, b, and c:", "[\nx = \frac{-(-78) \pm \sqrt{(-78)^2 - 4(4)(108)}}{2(4)}\n]", "Simplify:", "[\nx = \frac{78 \pm \sqrt{6084 - 1728}}{8}\n]", "[\nx = \frac{78 \pm \sqrt{4356}}{8}\n]", "Note that:\n√4356 = 66 (since 66 × 66 = 4356)", "So:", "[\nx = \frac{78 \pm 66}{8}\n]", "---", "### 3. Calculate the Two Roots", "First root:", "[\nx = \frac{78 + 66}{8} = \frac{144}{8} = 18\n]", "Second root:", "[\nx = \frac{78 - 66}{8} = \frac{12}{8} = \frac{3}{2} = 1.5\n]", "---", "## The Roots: x = 18 and x = 1.5", "The solutions to the equation 4x² – 78x + 108 = 0 are:", "- x = 18\n- x = 1.5", "These are rational roots, meaning they are simple fractions or whole numbers — useful in modeling real-life scenarios.", "---", "## Verifying the Roots", "Plugging x = 18 into the original equation:", "[\n4(18)^2 – 78(18) + 108 = 4(324) – 1404 + 108 = 1296 – 1404 + 108 = 0 \quad ✅\n]", "Plugging x = 1.5:", "[\n4(1.5)^2 – 78(1.5) + 108 = 4(2.25) – 117 + 108 = 9 – 117 + 108 = 0 \quad ✅\n]", "Both roots satisfy the equation.", "---", "## Real-World Applications", "Quadratic equations like 4x² – 78x + 108 = 0 appear in diverse fields:\n- Physics: Modeling projectile motion and quadratic motion under gravity\n- Engineering: Designing parabolic structures and optimizing material usage\n- Economics: Profit maximization where revenue and cost curves intersect\n- Geometry: Finding dimensions given area and side relationships", "---", "## Why Use the Quadratic Formula?", "- Handles all quadratic cases, including irrational and complex roots\n- Provides exact solutions without guesswork\n- Useful when factoring is difficult or impossible", "---", "## Conclusion", "Solving 4x² – 78x + 108 = 0 demonstrates the power of the quadratic formula in finding precise roots. With roots x = 18 and x = 1.5, this equation models scenarios where balance points or optimal values exist. Whether in academic settings or real-world problem-solving, mastering these techniques builds a strong foundation in algebra and beyond.", "---", "## Additional Tips", "- Always simplify radicals — 66 is irrational, not whole\n- Check discriminant (b²–4ac) to confirm root nature (positive = two real roots)\n- Practice factoring when possible for insight into equation structure", "---", "Keywords: quadratic equation solution, solve 4x² – 78x + 108 = 0, quadratic formula, algebra tutorial, real roots calculator, 4x²–78x+108 solutions, parabola intersections", "---", "Meta Description:\nLearn how to solve 4x² – 78x + 108 = 0 using the quadratic formula. Step-by-step explanation of roots, verification, and real-world applications with exact solutions and helpful tips."]