Discriminant: \(b^2 - 4ac = (-3)^2 - 4 \times 2 \times (-5) = 9 + 40 = 49\).
["Understanding the Discriminant: (b^2 - 4ac = 49) in Quadratic Equations", "When solving quadratic equations of the form (ax^2 + bx + c = 0), one of the most critical early steps is computing the discriminant. For the equation (2x^2 - 3x - 5 = 0), the discriminant is calculated as:", "[\n\Delta = b^2 - 4ac\n]", "Substituting (a = 2), (b = -3), and (c = -5):", "[\n\Delta = (-3)^2 - 4 \ imes 2 \ imes (-5) = 9 + 40 = 49\n]", "### Why Is the Discriminant Important?", "The discriminant tells you everything about the nature and number of solutions to a quadratic equation:", "- Positive discriminant ((\Delta > 0)): The equation has two distinct real roots.\n- Zero discriminant ((\Delta = 0)): There is exactly one real root (a repeated root).\n- Negative discriminant ((\Delta < 0)): The roots are complex (non-real).", "In our example, since (\Delta = 49 > 0), the equation (2x^2 - 3x - 5 = 0) has two distinct real solutions.", "### Breaking Down the Calculation", "Let’s walk through the math step-by-step:", "- Step 1: Square the coefficient (b):\n ((-3)^2 = 9)", "- Step 2: Compute the product (4ac):\n (4 \ imes 2 \ imes (-5) = -40)\n Since discriminant uses (-4ac), this becomes:\n (-4 \ imes 2 \ imes (-5) = +40)", "- Step 3: Combine the results:\n [\n \Delta = 9 + 40 = 49\n ]", "This positive value confirms two real roots — a key insight before solving further using the quadratic formula.", "### How to Use the Discriminant in Practice", "Knowing the discriminant allows you to:", "- Predict solution types without solving fully.\n- Estimate the behavior of parabolas in graphing.\n- Decide whether factoring or alternative methods are most efficient.", "For this equation, because (\Delta = 49 > 0), applicants or students can confidently state the equation has two distinct real solutions, and proceed to solving via:", "[\nx = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{3 \pm 7}{4}\n]", "Thus, (x = \frac{10}{4} = \frac{5}{2}) and (x = \frac{-4}{4} = -1).", "### Final Thoughts", "The discriminant is a powerful algebraic shortcut hidden in the coefficients of a quadratic equation. With (b^2 - 4ac = 49), you immediately know the solutions exist and are real and different — a crucial piece of information for both theoretical understanding and practical applications.", "---", "Keywords: discriminant, quadratic equation, (b^2 - 4ac), real roots, two distinct roots, solving quadratics, algebra tutorial, quadratic formula, discriminant interpretation", "Meta Description: Learn why (b^2 - 4ac = 49) for (2x^2 - 3x - 5 = 0) signals two distinct real solutions. Discover the meaning, calculation, and significance of the discriminant in quadratic equations."]
