Calculate the discriminant: \( b^2 - 4ac = (-12)^2 - 4 \times 3 \times 9 = 144 - 108 = 36 \).
["Calculate the Discriminant: A Step-by-Step Guide with Example", "The discriminant is a powerful tool in algebra, especially when analyzing quadratic equations. Understanding how to calculate it helps determine the nature of a quadratic equation’s roots—whether they are real and distinct, real and repeated, or complex. This article explains what the discriminant is, how to compute it, and walks you through a practical example with step-by-step calculations.", "---", "### What is the Discriminant?", "In a standard quadratic equation of the form:\n[\nax^2 + bx + c = 0\n]\nthe discriminant is given by the formula:\n[\nD = b^2 - 4ac\n]\nThe discriminant reveals critical information about the roots of the equation:\n- If ( D > 0 ): Two distinct real roots.\n- If ( D = 0 ): Exactly one real root (also called a repeated or double root).\n- If ( D < 0 ): Two complex conjugate roots.", "Calculating the discriminant is quick and essential in solving quadratic equations efficiently.", "---", "### How to Calculate the Discriminant: A Detailed Example", "Let’s compute the discriminant for the quadratic expression with coefficients:\n[\na = 3,\ b = -12,\ c = 9\n]", "Plug these values into the discriminant formula:\n[\nb^2 - 4ac = (-12)^2 - 4 \ imes 3 \ imes 9\n]", "Step-by-step breakdown:\n1. Compute ( (-12)^2 = 144 )\n2. Calculate ( 4 \ imes 3 \ imes 9 = 108 )\n3. Subtract: ( 144 - 108 = 36 )", "So, the discriminant is:\n[\nb^2 - 4ac = 36\n]", "---", "### What Does This Mean?", "Since the discriminant equals 36 (( D > 0 )), we conclude that the quadratic equation has two distinct real roots. This result confirms the quadratic has two different intercepts on the x-axis when graphed.", "---", "### Why Is Calculating the Discriminant Useful?", "- Quick Root Analysis: Avoids the need for full quadratic formula solving when only root nature is needed.\n- Equation Classification: Helps determine whether to expect real or complex roots.\n- Optimization in Design and Engineering: Used in modeling parabolic paths, structural designs, and economic forecasts.", "---", "### Conclusion", "Calculating the discriminant is a fundamental skill in algebra. With the example ( b^2 - 4ac = (-12)^2 - 4 \ imes 3 \ imes 9 = 144 - 108 = 36 ), we determined the discriminant is 36—indicating two real, distinct solutions. Mastering this calculation empowers you to analyze quadratic equations efficiently and confidently.", "---", "Keywords: discriminant calculation, quadratic equation, b squared minus 4ac, quadratic formula, real roots, discriminant example, mathematics tutorial, algebra help.\nMeta Description: Learn how to calculate the discriminant ( b^2 - 4ac ) with a step-by-step example. Understand root nature and equation classification using this key algebraic concept."]
