If the discriminant of a quadratic equation is negative, it indicates that the equation has no real solutions. In the context of a standard quadratic equation of the form ax2 + bx + c = 0, the discriminant is calculated using the formula D = b2 – 4ac.
When the discriminant (D) is negative (D < 0), this means that the term b2 is less than 4ac. In graphical terms, this implies that the parabola represented by the quadratic function does not intersect the x-axis. Instead, it either opens upwards or downwards (depending on the value of a) and exists entirely above or below the x-axis.
As a result, the solutions to the quadratic equation are considered complex (or imaginary) numbers. These solutions can be expressed using the formula for the roots:
- x = (-b ± √D) / (2a)
Since the square root of a negative number results in an imaginary unit (i), the roots can be rewritten as:
- x = (-b ± i√(-D)) / (2a)
In summary, when you encounter a negative discriminant, you can conclude that:
- The quadratic equation has no real solutions.
- It has two complex solutions that are conjugates of each other.
- The graph of the quadratic function does not cross the x-axis.
This characteristic is crucial in various applications, particularly in algebra and calculus, as it helps in understanding the nature of quadratic functions and their behaviors in different scenarios.