Calculate the discriminant: \( b^2 - 4ac = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64 \).
["# How to Calculate the Discriminant: A Complete Step-by-Step Guide (Including ( b^2 - 4ac = 64 ))", "The discriminant is a fundamental concept in algebra, especially when solving quadratic equations of the form ( ax^2 + bx + c = 0 ). Understanding it helps determine the nature of the roots—whether they are real or complex, and whether they are distinct or repeated. In this article, we’ll learn how to compute the discriminant using the formula ( b^2 - 4ac ), with a practical example showing that ( (-4)^2 - 4 \ imes 2 \ imes (-6) = 64 ).", "---", "## Why Calculate the Discriminant?", "The discriminant, denoted by ( D = b^2 - 4ac ), reveals crucial information about the solutions of a quadratic equation:", "- If ( D > 0 ): Two distinct real roots\n- If ( D = 0 ): One real double root (repeated root)\n- If ( D < 0 ): Two complex conjugate roots", "This makes the discriminant a powerful tool in algebra and calculus for predicting the nature of solutions without fully solving the equation.", "---", "## The Discriminant Formula", "For a quadratic equation ( ax^2 + bx + c = 0 ), the discriminant is calculated as:", "[\nD = b^2 - 4ac\n]", "This formula appears in the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{D}}{2a}\n]", "---", "## Example: Calculate the Discriminant of ( 2x^2 - 4x - 6 = 0 )", "Let’s apply the discriminant calculation step-by-step using the given equation:", "[\n2x^2 - 4x - 6 = 0\n]", "Here, the coefficients are:\n- ( a = 2 )\n- ( b = -4 )\n- ( c = -6 )", "Now, substitute into the discriminant formula:", "[\nb^2 - 4ac = (-4)^2 - 4 \ imes 2 \ imes (-6)\n]", "Break it down:", "- ( (-4)^2 = 16 )\n- ( 4 \ imes 2 \ imes (-6) = -48 ), so ( -4ac = -4 \ imes 2 \ imes (-6) = +48 )", "Now add:", "[\n16 + 48 = 64\n]", "Therefore, the discriminant is:", "[\nD = 64\n]", "---", "## What Does ( D = 64 ) Mean?", "Since ( D = 64 > 0 ), the quadratic equation has two distinct real roots. This confirms there are no complex solutions, and the roots will be different real numbers.", "---", "## Summary", "- The discriminant ( b^2 - 4ac ) determines the nature of quadratic equation solutions.\n- In our example, ( D = (-4)^2 - 4 \ imes 2 \ imes (-6) = 64 ).\n- A positive discriminant confirms two distinct real roots.\n- Using the discriminant saves time by predicting solution types without fully solving the equation.", "---", "Whether you're solving equations for homework or deeper mathematical analysis, mastering the discriminant is essential. Start calculating ( b^2 - 4ac ) today and unlock insight into quadratic behavior!", "---", "Keywords: discriminant, quadratic equation, ( b^2 - 4ac ), solve quadratic, mathematical formula, real roots, complex roots, algebra, quadratic formula."]
