Understanding the Nature of Quadratic Solutions: Explaining Negative Discriminant and Complex Solutions

If b²-4ac < 0

If b²-4ac is less than zero, it means that the discriminant (b²-4ac) of a quadratic equation is negative

If b²-4ac is less than zero, it means that the discriminant (b²-4ac) of a quadratic equation is negative. The discriminant is calculated as part of the quadratic formula and indicates the nature of the solutions for a quadratic equation of the form ax² + bx + c = 0.

When the discriminant is negative, it means that the quadratic equation does not have any real solutions. Instead, it has two complex solutions (complex conjugates) because the square root of a negative number is imaginary.

To further explain this concept, let’s break it down into three cases based on the value of the discriminant:

1. b²-4ac > 0: When the discriminant is positive, it indicates that the quadratic equation has two distinct real solutions. This occurs when the graph of the equation intersects the x-axis at two different points.

2. b²-4ac = 0: When the discriminant is zero, it means that the quadratic equation has a single real solution. This occurs when the graph of the equation touches the x-axis at a single point.

3. b²-4ac < 0: When the discriminant is negative, as mentioned in the original question, it signifies that the quadratic equation has no real solutions. Instead, it has two complex solutions. This occurs when the graph of the equation does not intersect or touch the x-axis. In summary, when b²-4ac is less than zero, the quadratic equation does not have any real solutions but rather two complex solutions.

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