4x² + 42x + 108 = 180
["Solving the Quadratic Equation: 4x² + 42x + 108 = 180", "When faced with a quadratic equation like 4x² + 42x + 108 = 180, solving for x might seem challenging at first, but with a clear step-by-step approach, it becomes manageable. This article guides you through solving this equation, explains key concepts, and provides useful tips for handling similar quadratic problems.", "---", "### Step 1: Simplify the Equation", "Start by bringing all terms to one side to form a standard quadratic equation equal to zero:", "$$\n4x² + 42x + 108 - 180 = 0\n$$", "Simplify the constants:", "$$\n4x² + 42x - 72 = 0\n$$", "---", "### Step 2: Simplify Further by Dividing by the GCF", "Check if all coefficients share a common factor. The coefficients 4, 42, and -72 share a GCF of 2. Divide every term by 2:", "$$\n2x² + 21x - 36 = 0\n$$", "Now we have a simpler quadratic equation ready for solving.", "---", "### Step 3: Use the Quadratic Formula", "Since factoring may not always be straightforward, use the quadratic formula:", "$$\nx = \frac{-b \pm \sqrt{b² - 4ac}}{2a}\n$$", "For the simplified equation 2x² + 21x - 36 = 0, identify:", "- ( a = 2 )\n- ( b = 21 )\n- ( c = -36 )", "---", "### Step 4: Calculate the Discriminant", "The discriminant ( \Delta = b² - 4ac ):", "$$\n\Delta = 21² - 4(2)(-36) = 441 + 288 = 729\n$$", "Since ( \Delta = 729 > 0 ), there are two distinct real solutions.", "---", "### Step 5: Plug into the Quadratic Formula", "$$\nx = \frac{-21 \pm \sqrt{729}}{2 \ imes 2} = \frac{-21 \pm 27}{4}\n$$", "This gives two possible solutions:", "- ( x = \frac{-21 + 27}{4} = \frac{6}{4} = \frac{3}{2} )\n- ( x = \frac{-21 - 27}{4} = \frac{-48}{4} = -12 )", "---", "### Step 6: Final Answer", "The solutions to the equation 4x² + 42x + 108 = 180 are:", "$$\n\boxed{x = \frac{3}{2} \quad \ ext{and} \quad x = -12}\n$$", "---", "### Why Knowing This Equation Matters", "Solving quadratics like this is essential in algebra, physics, engineering, and economics, where parabolic relationships model real-world phenomena such as projectile trajectories, profit maximization, and optimization problems.", "---", "### Tips for Solving Quadratic Equations Easily", "- Always simplify the equation to standard form before applying formulas.\n- Factor first when possible, but use the quadratic formula when needed.\n- Check your answer by plugging values back into the original equation.\n- Understand how the discriminant determines the nature of the roots (real and distinct, real and repeated, or complex).", "---", "By following these clear steps and practicing regularly, solving quadratic equations becomes not just a math task—but a valuable problem-solving skill. Whether you're a student, teacher, or a curious learner, mastering quadratics opens the door to deeper mathematical understanding!"]
