Understanding the Discriminant

• The discriminant is the expression under the square root in the quadratic formula: b² – 4ac.
• It provides valuable information about the nature of the solutions to a quadratic equation.

Interpreting the Discriminant

• Positive Discriminant (b² – 4ac > 0): The quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
• Zero Discriminant (b² – 4ac = 0): The quadratic equation has one repeated real root. This means the parabola touches the x-axis at exactly one point.
• Negative Discriminant (b² – 4ac < 0): The quadratic equation has no real roots. This means the parabola does not intersect the x-axis. The solutions are complex numbers.

Example

Consider the quadratic equation: 2x² – 5x + 3 = 0

• Calculate the discriminant: b² – 4ac = (-5)² – 4(2)(3) = 1
• Since the discriminant is positive, the equation has two distinct real roots.

Key Points

• The discriminant is a powerful tool for analyzing quadratic equations.
• It helps to determine the number and nature of the solutions without actually solving the equation.
• Understanding the discriminant can save time and effort when solving quadratic equations.

Learn with an example