Discriminant \( = b^2 - 4ac = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64 \).
["Discriminant Formula Explained: ( b^2 - 4ac = 64 )", "When solving quadratic equations of the form ( ax^2 + bx + c = 0 ), the discriminant plays a crucial role in determining the nature of the roots. For any quadratic equation, the discriminant is calculated using the formula:\n[ D = b^2 - 4ac ]", "Understanding the discriminant allows students, educators, and math enthusiasts to quickly identify whether solutions are real and distinct, real and repeated, or complex—all based on the value of ( D ).", "---", "### What Is the Discriminant?", "The discriminant ( D = b^2 - 4ac ) reveals key characteristics of the roots without fully solving the equation:", "- If ( D > 0 ): Two distinct real roots exist.\n- If ( D = 0 ): Exactly one real root (a repeated or double root).\n- If ( D < 0 ): The roots are complex (non-real).", "---", "### Applying the Discriminant to ( (-4)^2 - 4 \ imes 2 \ imes (-6) = 64 )", "Let’s analyze a specific example: the quadratic expression with coefficients ( a = 2 ), ( b = -4 ), and ( c = -6 ). (Note: ( (-4)^2 = 16 ) and ( -4 \ imes 2 \ imes (-6) = 48 ), so indeed:)", "[\nD = (-4)^2 - 4 \ imes 2 \ imes (-6)\n]", "Step-by-step:", "1. Compute ( b^2 ):\n [\n (-4)^2 = 16\n ]\n2. Calculate the product term:\n [\n 4ac = 4 \ imes 2 \ imes (-6) = -48\n ]\n3. Plug values into the discriminant formula:\n [\n D = 16 - (-48) = 16 + 48 = 64\n ]", "Since ( D = 64 > 0 ), this quadratic equation has two distinct real roots, confirming the equation will intersect the x-axis at two points.", "---", "### Why Is the Discriminant Important?", "- Quick root analysis: No need to solve the entire equation—just compute ( D ) to classify roots.\n- Guides graph sketching: Determines whether a parabola opens upward/downward and how many times it crosses the x-axis.\n- Versatile application: Useful in physics, engineering, economics, and many areas where quadratic models describe real-world phenomena.", "---", "### How to Use This in Real Problems", "Suppose you're solving a physics problem modeling projectile motion using a quadratic equation, and you obtain ( 2x^2 - 4x - 6 = 0 ). By computing the discriminant:", "[\nD = (-4)^2 - 4 \ imes 2 \ imes (-6) = 16 + 48 = 64\n]", "You immediately know there are two distinct real solutions—helpful for predicting two distinct landing points or time intervals.", "---", "### Summary", "The discriminant ( D = b^2 - 4ac ) is a powerful tool that simplifies quadratic analysis:", "- For our example, ( D = 64 ) → Two real, distinct roots ✅\n- It tells you how many times a quadratic function crosses the x-axis without completing factoring or applying the quadratic formula fully.", "Whether in algebra, calculus, or applied sciences, mastering the discriminant is essential for efficient and insightful problem-solving.", "---", "Keywords: discriminant formula, discriminant ( D = b^2 - 4ac ), quadratic equation roots, real and complex roots, mathematics teaching, algebra guide, solving quadratics", "Meta Description:\nLearn how the discriminant ( b^2 - 4ac = 64 ) determines root types—perfect for mastering quadratic equations efficiently with clear calculation examples and real-world applications."]
