\[ b^2 - 4ac = (-12)^2 - 4 \times 3 \times 9 = 144 - 108 = 36 \]

["Understanding the Discriminant: b² - 4ac and Its Calculation Using (-12)² – 4×3×9", "When solving quadratic equations of the form ( ax^2 + bx + c = 0 ), one key mathematical expression is the discriminant, defined as ( b^2 - 4ac ). This value determines the nature of the roots—whether they are real and distinct, real and equal, or complex.", "In this article, we explore the discriminant calculation step-by-step using the numbers ( b = -12 ), ( a = 3 ), and ( c = 9 ). The expression simplifies as follows:", "[\nb^2 - 4ac = (-12)^2 - 4 \ imes 3 \ imes 9\n]", "First, compute ( (-12)^2 ):\n[\n(-12)^2 = 144\n]", "Next, calculate ( 4 \ imes 3 \ imes 9 ):\n[\n4 \ imes 3 = 12,\quad 12 \ imes 9 = 108\n]", "Now, subtract:\n[\n144 - 108 = 36\n]", "Thus,\n[\nb^2 - 4ac = 36\n]", "### What does a discriminant of 36 mean?", "Since the discriminant is positive (( 36 > 0 )), the quadratic equation has two distinct real roots. This result helps us quickly assess the behavior of the parabola represented by the equation—specifically, that it crosses the x-axis at two points.", "### Summary", "- The discriminant ( b^2 - 4ac ) reveals important information about quadratic roots.\n- For ( a = 3 ), ( b = -12 ), and ( c = 9 ), we compute ( b^2 - 4ac = 36 ).\n- A positive discriminant means two distinct real solutions.", "This straightforward computation makes it easy to evaluate quadratic expressions without always factoring or applying the quadratic formula—ideal for both students and educators seeking clarity on this fundamental algebra concept.", "#### Further reading:\n- Step-by-step guide to solving quadratic equations\n- Understanding each root type through the discriminant\n- Applications of the quadratic formula in real-world problems"]