\[ b^2 - 4ac = (-5)^2 - 4 imes 2 imes (-3) = 25 + 24 = 49 \]

["Understanding the Discriminant: Solving Quadratic Equations with the Formula ( b^2 - 4ac )", "The discriminant of a quadratic equation is a critical mathematical tool that determines the nature and number of solutions a quadratic equation ( ax^2 + bx + c = 0 ) has. It is calculated using the formula:", "[\n\Delta = b^2 - 4ac\n]", "In this specific example, we explore the discriminant through a concrete computation:", "[\n\Delta = (-5)^2 - 4 \ imes 2 \ imes (-3)\n]", "### Step-by-Step Breakdown of the Computation", "First, compute each component of the discriminant:", "- First term: ( (-5)^2 = 25 )\n- Second term: ( 4 \ imes 2 \ imes (-3) = 8 \ imes (-3) = -24 )", "Now plug these into the discriminant formula:", "[\n\Delta = 25 - (-24) = 25 + 24 = 49\n]", "Thus, ( b^2 - 4ac = 49 ).", "### What Does the Discriminant Tell Us?", "Since ( \Delta = 49 ) is positive and a perfect square, the quadratic equation has two distinct real and rational roots. This outcome means the quadratic crosses the x-axis at two points.", "### Solving the Example Quadratic Equation", "Using the values ( a = 2 ), ( b = -5 ), and ( c = -3 ), the equation becomes:", "[\n2x^2 - 5x - 3 = 0\n]", "Apply the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-5) \pm \sqrt{49}}{2 \ imes 2} = \frac{5 \pm 7}{4}\n]", "This gives two solutions:", "- ( x_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3 )\n- ( x_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2} )", "### Why Is the Discriminant So Important?", "- Nature of Roots:\n - ( \Delta > 0 ): Two distinct real solutions\n - ( \Delta = 0 ): One real repeated root\n - ( \Delta < 0 ): Two complex conjugate roots\n- Graphical Insight: A larger positive discriminant indicates wider separation of the parabola’s roots on the x-axis.", "### Summary", "In this example, evaluating ( b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot (-3) = 49 ) reveals not only the deterministic outcome of two real roots but also serves as a foundation for deeper algebraic analysis and graphing behavior. Mastering this calculation empowers students and practitioners alike to quickly assess the behavior of quadratic equations without solving them fully.", "---", "Keytakeaways:\n- Use ( b^2 - 4ac ) to determine solution types.\n- Accurate arithmetic prevents errors in discriminant calculation.\n- Real-world applications include physics, engineering, and optimization problems.", "---", "Optimize your understanding of quadratics by routinely practicing discriminant calculations—your path to mastering algebra begins here!"]