A quadratic equation is typically expressed in the standard form:
ax² + bx + c = 0
where a, b, and c are constants, and a cannot be zero. The number of real solutions for a quadratic equation is determined by its discriminant, denoted as D.
The discriminant is calculated using the formula:
D = b² – 4ac
Now, this discriminant gives us valuable insight into the nature of the solutions:
- If D > 0, the quadratic equation has two distinct real solutions.
- If D = 0, the quadratic equation has exactly one real solution, also known as a repeated or double root.
- If D < 0, the solutions are complex or imaginary, meaning there are no real solutions.
So when we say that a quadratic equation has exactly one real number solution, we are essentially stating that the discriminant equals zero (D = 0). This means that the graph of the quadratic, which is a parabola, touches the x-axis at exactly one point. In simpler terms, the parabola has a vertex that lies on the x-axis, indicating that the single solution is the x-coordinate of the vertex.
In summary, the discriminant plays a crucial role in determining the nature of the roots of a quadratic equation, and when it is zero, we can confidently say that there is one real solution.