When dealing with quadratic equations, which are typically expressed in the standard form ax2 + bx + c = 0, the discriminant plays a crucial role in determining the nature of the solutions. The discriminant is calculated using the formula D = b2 – 4ac.
If the discriminant (D) equals zero, it indicates that the quadratic equation has exactly one real solution. This particular situation is known as a perfect square trinomial. Geometrically, this means that the graph of the quadratic function touches the x-axis at a single point, known as the vertex, instead of crossing it.
The real solution can be calculated using the quadratic formula: x = rac{{-b}}{{2a}}. Here, the solution represents the x-coordinate where the vertex is located, signifying that the quadratic has a double root.
In summary, if the discriminant of a quadratic equation is zero, the equation has precisely one real solution, highlighting a unique scenario where the parabola remains tangent to the x-axis.