The graph of a quadratic reveals valuable information about its solutions. Two real solutions correspond to distinct x-intercepts, two imaginary solutions lead to a parabola above or below the x-axis, and one real solution results in a tangent point with a discriminant of zero.
The graph of a quadratic equation, expressed in the form y = ax^2 + bx + c, provides insights into the nature of its solutions. For a quadratic with two real solutions, the graph intersects the x-axis at two distinct points. These points, called the roots, represent the values of x where the quadratic equation equals zero. In this case, the parabola opens either upwards (if a > 0) or downwards (if a < 0).
When a quadratic equation has two imaginary solutions, its graph does not intersect the x-axis. Instead, the entire parabola lies above or below the x-axis, depending on the sign of the leading coefficient. The absence of real roots indicates that the quadratic crosses the x-axis at points with complex coordinates.
For a quadratic with one real solution, the graph touches the x-axis at a single point. This situation occurs when the discriminant (b^2 - 4ac) is equal to zero, resulting in a perfect square trinomial and a repeated root. The parabola just grazes the x-axis without crossing it, forming a tangent point.