\[ b^2 - 4ac = (-4)^2 - 4 imes 2 imes (-6) = 16 + 48 = 64 \]

["Understanding the Quadratic Formula: A Step-by-Step Breakdown of ( b^2 - 4ac = (-4)^2 - 4 \cdot 2 \cdot (-6) )", "If you’ve ever worked with quadratic equations, you’ve likely encountered the discriminant — the expression ( b^2 - 4ac ) that determines the nature of the roots. What happens when you compute this deceptively simple formula? Let’s take a deep dive into an example that reveals its power:\n[ b^2 - 4ac = (-4)^2 - 4 \ imes 2 \ imes (-6) = 16 + 48 = 64 ]", "### What Is the Discriminant?", "In the standard quadratic equation:\n[ ax^2 + bx + c = 0 ]\nthe discriminant is given by ( D = b^2 - 4ac ). This value tells us critical information about the equation’s solutions:\n- If ( D > 0 ): Two distinct real roots\n- If ( D = 0 ): One real double root\n- If ( D < 0 ): Two complex roots", "This formula is foundational in algebra and calculus, forming the backbone of solving quadratic systems across physics, engineering, and data modeling.", "### Breaking Down the Computation Step-by-Step", "Let’s unpack the specific calculation:\n[ (-4)^2 - 4 \ imes 2 \ imes (-6) ]", "1. First Term: Squaring ( -4 )\n( (-4)^2 = 16 )", "2. Second Term: Multiplication in the Discriminant\n( 4 \ imes 2 \ imes (-6) = 8 \ imes (-6) = -48 )\nBut note the sign in the discriminant: it's minus this product, so:\n( -4ac = - (4 \ imes 2 \ imes -6) = - (-48) = +48 )", "3. Final Calculation:\nTogether:\n( b^2 - 4ac = 16 + 48 = 64 )", "Thus, the discriminant equals 64 — a positive number, meaning the quadratic equation has two distinct real solutions. This makes the equation fully solvable over the real numbers.", "### Why This Matters in Real-World Applications", "Understanding how the discriminant behaves not only streamlines algebraic computation but also enhances problem-solving in applied fields. For example:\n- In physics, determining whether a projectile hits the ground at two points.\n- In optimization, identifying the number and nature of critical points.\n- In engineering, analyzing stability by studying system response equations.", "The simple act of calculating ( b^2 - 4ac ) opens the door to insightful conclusions about the behavior of quadratic systems without needing to solve for roots explicitly.", "### Conclusion", "The computation ( (-4)^2 - 4 \cdot 2 \cdot (-6) = 64 ) exemplifies how basic algebraic manipulation uncovers deeper mathematical truths about quadratic equations. By mastering the discriminant’s logic and arithmetic, learners gain a powerful tool to predict, analyze, and solve quadratic phenomena—making algebra not just a set of formulas, but a dynamic pathway to understanding continuity, variation, and real-world dynamics.", "So next time you see this expression, remember: it’s more than numbers — it’s the key to unlocking the behavior of quadratic relationships."]