If the parabola's vertex is on the \( x \)-axis, then the corresponding equation has a single repeated root on the line of symmetry, and this distance term is zero, algebraically, the (iii) commonly known as \( b^{2}-4 a c=0 \)

To analyze the condition for a parabola with its vertex on the \( x \)-axis and a single repeated root, we can start by recalling the standard form of a quadratic equation: \[ y = ax^2 + bx + c \] The vertex of this parabola is given by the formula: \[ x_v = -\frac{b}{2a} \] For the vertex to be on the \( x \)-axis, the \( y \)-coordinate of the vertex must be zero. The \( y \)-coordinate of the vertex can be found by substituting \( x_v \) back into the equation: \[ y_v = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c \] Simplifying this expression gives: \[ y_v = a\left(\frac{b^2}{4a^2}\right) - \frac{b^2}{2a} + c \] \[ y_v = \frac{b^2}{4a} - \frac{2b^2}{4a} + c \] \[ y_v = -\frac{b^2}{4a} + c \] Setting \( y_v = 0 \) for the vertex to be on the \( x \)-axis, we have: \[ -\frac{b^2}{4a} + c = 0 \] Rearranging this gives: \[ c = \frac{b^2}{4a} \] Now, substituting \( c \) back into the quadratic equation's discriminant condition for a single repeated root, we have: \[ b^2 - 4ac = 0 \] Substituting \( c \): \[ b^2 - 4a\left(\frac{b^2}{4a}\right) = 0 \] \[ b^2 - b^2 = 0 \] This confirms that the condition \( b^2 - 4ac = 0 \) holds true when the vertex is on the \( x \)-axis, indicating a single repeated root. Thus, the conclusion is that for a parabola with its vertex on the \( x \)-axis, the discriminant condition \( b^2 - 4ac = 0 \) is satisfied, confirming the presence of a single repeated root.