Understanding the Nature of Solutions: Exploring Real and Complex Roots in Quadratic Equations

If b²-4ac < 0

If the expression b² – 4ac is less than zero, it indicates that the discriminant (the value under the square root in the quadratic formula) is negative

If the expression b² – 4ac is less than zero, it indicates that the discriminant (the value under the square root in the quadratic formula) is negative. In other words, it means that the quadratic equation ax² + bx + c = 0 has no real solutions.

This condition can be used to determine the nature of the solutions for a quadratic equation. There are three possible cases:

1. If b² – 4ac > 0, then the discriminant is positive, indicating that the equation has two distinct real solutions.
2. If b² – 4ac = 0, then the discriminant is zero, indicating that the equation has one real solution (also called a repeated root or double root).
3. If b² – 4ac < 0, then the discriminant is negative, indicating that the equation has no real solutions (only complex solutions). Complex solutions involve the use of imaginary numbers (numbers involving the square root of -1, denoted as "i"). These solutions will have the form of a + bi, where "a" and "b" are real numbers. For example, let's consider the quadratic equation x² + 2x + 5 = 0. The coefficients are a = 1, b = 2, and c = 5. Calculating the discriminant: b² - 4ac = (2)² - 4(1)(5) = 4 - 20 = -16 Since the discriminant is negative (-16 < 0), this equation has no real solutions. The solutions will be complex. To find the complex solutions, we can rewrite the equation as: x = (-b ± √(b² - 4ac)) / (2a) x = (-2 ± √(-16)) / (2) x = (-2 ± 4i) / 2 Simplifying further: x = -1 ± 2i Therefore, the complex solutions for this quadratic equation are x = -1 + 2i and x = -1 - 2i. In summary, when b² - 4ac < 0, it means that the quadratic equation has no real solutions and the solutions are complex.

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