A quadratic equation can be expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to this equation can be determined using the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)
The term under the square root, b² – 4ac, is known as the discriminant. The value of the discriminant determines the nature of the roots of the quadratic equation:
- If the discriminant is positive (b² – 4ac > 0), there are two distinct real solutions.
- If the discriminant is zero (b² – 4ac = 0), there is exactly one real solution (also called a repeated root).
- If the discriminant is negative (b² – 4ac < 0), there are no real solutions, only complex solutions.
Therefore, if a quadratic equation has exactly one real number solution, the discriminant must be equal to zero. This means that the equation touches the x-axis at exactly one point, resulting in one real solution.