The real solutions of a quadratic equation correspond to the points where the graph of the quadratic function intersects the x-axis. This is significant because these intersections indicate the values of x for which the function equals zero, effectively solving the equation.
A quadratic equation can be expressed in the standard form as:
ax² + bx + c = 0
Here, ‘a’, ‘b’, and ‘c’ are constants, and ‘a’ should not equal zero. The graph of the quadratic function, which is represented by:
f(x) = ax² + bx + c
, has a characteristic U-shape known as a parabola. The orientation of this parabola depends on the value of ‘a’: if ‘a’ is positive, the parabola opens upwards, and if ‘a’ is negative, it opens downwards.
When we find the real solutions of the quadratic equation, we are essentially locating the x-coordinates at which the parabola crosses the x-axis. If the graph touches the x-axis at a single point (a vertex), the quadratic equation has one real solution (or a repeated root). Alternatively, if the graph crosses the x-axis at two distinct points, there are two real solutions. If the graph does not intersect the x-axis at all, the quadratic function has no real solutions (when the discriminant, b² – 4ac, is negative).
In summary, the relationship between the real solutions of a quadratic equation and the graph of its corresponding function is visually represented by the intersection points with the x-axis, which also tells us important information about the nature of the solutions.