Discriminant: \( (-3)^2 - 4 \times 2 \times (-5) = 9 + 40 = 49 \).
["Understanding the Discriminant: Solving Quadratic Equations with ( (-3)^2 - 4 \ imes 2 \ imes (-5) = 49 )", "Solving quadratic equations is a fundamental skill in algebra, and one of the key elements in determining the nature of the roots is the discriminant. For a quadratic equation in the standard form ( ax^2 + bx + c = 0 ), the discriminant is calculated using the formula:", "[\n\Delta = b^2 - 4ac\n]", "This value reveals important information about the number and type of roots—whether they are real and distinct, real and repeated, or complex.", "---", "### What is the Discriminant for the Equation with Discriminant Calculated as ( (-3)^2 - 4 \ imes 2 \ imes (-5) = 49 )?", "Let’s break down the components:", "- Coefficients:\n - ( a = 2 )\n - ( b = -3 )\n - ( c = -5 )", "Using the discriminant formula:", "[\n\Delta = (-3)^2 - 4 \ imes 2 \ imes (-5)\n]", "Step 1: Square ( b ):\n[\n(-3)^2 = 9\n]", "Step 2: Compute ( -4ac ):\n[\n-4 \ imes 2 \ imes (-5) = -8 \ imes (-5) = 40\n]", "Step 3: Add both parts:\n[\n\Delta = 9 + 40 = 49\n]", "---", "### Interpreting the Discriminant Value – What Does ( \Delta = 49 ) Mean?", "Since the discriminant is 49—a positive perfect square—the quadratic equation has:", "- Two distinct real roots\n- The solutions are rational numbers because the square root of 49 is 7, a whole number.", "This makes the quadratic equation have two different real solutions, which can be found easily using the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{3 \pm 7}{4}\n]", "Calculating:", "- ( x_1 = \frac{3 + 7}{4} = \frac{10}{4} = 2.5 )\n- ( x_2 = \frac{3 - 7}{4} = \frac{-4}{4} = -1 )", "Thus, the roots are ( x = 2.5 ) and ( x = -1 ).", "---", "### Why the Discriminant Is Essential in Algebra", "The discriminant is a powerful tool because:", "- It determines root types without fully solving the equation\n- A positive ( \Delta ) → two distinct real roots\n- Zero ( \Delta ) → one repeated real root\n- Negative ( \Delta ) → complex conjugate roots", "Understanding and calculating the discriminant helps students and mathematicians quickly assess quadratic behavior, saving time in analysis and problem-solving.", "---", "### Conclusion", "In the example ( (-3)^2 - 4 \ imes 2 \ imes (-5) = 49 ), the discriminant equals 49, confirming two distinct real solutions. This demonstrates how the discriminant acts as a shortcut to understanding quadratic equations’ nature — a cornerstone of algebra. Whether you're solving problems manually or using technology, mastering the discriminant is key to efficiently analyzing quadratic functions.", "Keywords: Discriminant, quadratic equation, ( (-3)^2 ), ( -4ac ), real roots, quadratic formula, algebra, mathematics teaching, solving quadratics, discriminant calculator.", "---", "Mastering the discriminant empowers you to quickly evaluate equations and ensure accurate, efficient solutions—essential for success in algebra and beyond."]
