Calculate the discriminant: \( b^2 - 4ac = (-3)^2 - 4(2)(-5) = 9 + 40 = 49 \).
["# Calculate the Discriminant: ( b^2 - 4ac = (-3)^2 - 4(2)(-5) = 49 )", "Understanding the discriminant is a crucial step in solving quadratic equations, and today we’ll walk through the full calculation of the discriminant using the standard form of a quadratic equation: ( ax^2 + bx + c = 0 ).", "## What Is the Discriminant?", "The discriminant is a key component in determining the nature of the roots of a quadratic equation. Given a quadratic in the form:", "[\nax^2 + bx + c = 0\n]", "The discriminant ( D ) is calculated using the formula:", "[\nD = b^2 - 4ac\n]", "The value of ( D ) tells us how many real and distinct roots the equation has:", "- If ( D > 0 ): Two distinct real roots.\n- If ( D = 0 ): One real (repeated) root.\n- If ( D < 0 ): Two complex conjugate roots.", "## Applying the Formula with Example Coefficients", "Consider the quadratic equation:", "[\n2x^2 - 3x - 5 = 0\n]", "Here, the coefficients are:\n- ( a = 2 )\n- ( b = -3 )\n- ( c = -5 )", "Now compute the discriminant step-by-step:", "[\nD = b^2 - 4ac = (-3)^2 - 4(2)(-5)\n]", "Calculate each term:", "- ( (-3)^2 = 9 )\n- ( 4 \ imes 2 \ imes (-5) = -40 ), but since we subtract a negative, this becomes ( -4 \ imes 2 \ imes (-5) = +40 )", "So,", "[\nD = 9 + 40 = 49\n]", "## Interpretation of the Result", "Since ( D = 49 > 0 ), this quadratic equation has two distinct real roots.", "To find the roots, use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4}\n]", "This gives:", "[\nx = \frac{3 + 7}{4} = \frac{10}{4} = 2.5 \quad \ ext{and} \quad x = \frac{3 - 7}{4} = \frac{-4}{4} = -1\n]", "## Why Calculating the Discriminant Matters", "Evaluating the discriminant before even solving the full equation saves time and confirms whether roots are real and unique. Whether you're a student learning algebra or a developer implementing math functions, computing ( b^2 - 4ac ) is a fast and reliable way to assess the behavior of quadratic equations.", "---", "Summary:\n- Given ( 2x^2 - 3x - 5 = 0 )\n- Discriminant: ( (-3)^2 - 4(2)(-5) = 9 + 40 = 49 )\n- Since ( 49 > 0 ), the equation has two distinct real roots.", "Calculate the discriminant with confidence—empower your math today!", "---", "Keywords:\ndiscriminant formula, quadratic equation discriminant, calculate discriminant, ( b^2 - 4ac ), real roots quadratic, math tutorial, quadratic formula application, determine root nature"]
