The quadratic formula defines the points $$ (x ; 0) $$ on the parabolic graph, where the parabola $$ y=a x^{2}+b x+c $$ crosses the $$ x $$-axis and it can be separated into two terms, $$ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} $$ $$ x=-\frac{b}{2 a} \pm \frac{\sqrt{b^{2}-4 a c}}{2 a} $$ The first term $$ -\frac{b}{2 a} $$ describes the (i) the line $$ x=-\frac{b}{2 a} $$. The second term $$ \frac{\sqrt{b^{2}-4 a c}}{2 a} $$, gives the (ii) 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)

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Let's break down the information provided in the problem step by step.

1. **Understanding the Quadratic Formula**:
   The quadratic formula is given as:
   $$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$
   This formula provides the $$ x $$-coordinates of the points where the parabola $$ y = ax^2 + bx + c $$ intersects the $$ x $$-axis.

2. **Separation into Two Terms**:
   The formula can be separated into two parts:
   - The first term:
     $$
     x = -\frac{b}{2a}
     $$
     This term represents the line of symmetry of the parabola, which is the vertical line that passes through the vertex of the parabola.

- The second term:
     $$
     x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}
     $$
     The second term, $$ \frac{\sqrt{b^2 - 4ac}}{2a} $$, represents the distance from the vertex to the points where the parabola intersects the $$ x $$-axis.

3. **Vertex on the $$ x $$-axis**:
   If the vertex of the parabola is on the $$ x $$-axis, it means that the parabola touches the $$ x $$-axis at that point, indicating a single repeated root. This occurs when the discriminant $$ b^2 - 4ac $$ is equal to zero.

- Algebraically, this means:
     $$
     b^2 - 4ac = 0
     $$
     Therefore, the distance term $$ \frac{\sqrt{b^2 - 4ac}}{2a} $$ becomes zero, leading to a single solution for $$ x $$:
     $$
     x = -\frac{b}{2a}
     $$

4. **Filling in the Blanks**:
   - (i) The line $$ x = -\frac{b}{2a} $$ describes the **line of symmetry** of the parabola.
   - (ii) The second term $$ \frac{\sqrt{b^2 - 4ac}}{2a} $$ gives the **distance from the vertex to the $$ x $$-intercepts**.
   - (iii) The algebraic condition for the vertex being on the $$ x $$-axis is that the **discriminant is zero**.

In summary:
- (i) line of symmetry
- (ii) distance from the vertex to the $$ x $$-intercepts
- (iii) discriminant is zero
- (i) line of symmetry - (ii) distance from the vertex to the $$ x $$-intercepts - (iii) discriminant is zero

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