SECTION C: SIGNIFICANCE OF THE QUADRATIC FORMULA [3] 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, $$ \begin{array}{l} x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \ x=-\frac{b}{2 a} \pm \frac{\sqrt{b^{2}-4 a c}}{2 a} \end{array} $$ The first term $$ -\frac{b}{2 a} $$ describes the (i) $$ \qquad $$ , the line $$ x=-\frac{b}{2 a} $$. The second term $$ \frac{\sqrt{b^{2}-4 a c}}{2 a} $$, gives the (ii) $$ \qquad $$ the roots are away from the axis of symmetry. 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 $$ b^{2}-4 a c=0 $$.

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To fill in the blanks in the provided text regarding the significance of the quadratic formula, let's analyze each part step by step.

1. **First Term $$ -\frac{b}{2a} $$**:
   - This term represents the x-coordinate of the vertex of the parabola. The vertex is the point where the parabola changes direction, and it is also the line of symmetry for the parabola.
   - Therefore, the first blank (i) can be filled with **"vertex"**.

2. **Second Term $$ \frac{\sqrt{b^{2}-4ac}}{2a} $$**:
   - This term represents the distance from the vertex to the points where the parabola intersects the x-axis (the roots). The roots are the solutions to the quadratic equation, and this term indicates how far the roots are from the line of symmetry (the vertex).
   - Thus, the second blank (ii) can be filled with **"distance from the vertex to the roots"**.

3. **Condition $$ b^{2}-4ac=0 $$**:
   - This condition indicates that the parabola touches the x-axis at exactly one point, meaning it has a single repeated root. This occurs when the vertex is on the x-axis, and the distance from the vertex to the roots is zero.
   - Therefore, the third blank (iii) can be filled with **"discriminant"**.

Putting it all together, the completed text would read:

The first term $$ -\frac{b}{2 a} $$ describes the (i) **vertex**, the line $$ x=-\frac{b}{2 a} $$. The second term $$ \frac{\sqrt{b^{2}-4 a c}}{2 a} $$, gives the (ii) **distance from the vertex to the roots** the roots are away from the axis of symmetry.

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 **discriminant** $$ b^{2}-4 a c=0 $$.
The first term $$ -\frac{b}{2a} $$ describes the **vertex**, the line $$ x=-\frac{b}{2a} $$. The second term $$ \frac{\sqrt{b^{2}-4ac}}{2a} $$ gives the **distance from the vertex to the roots**, indicating how far the roots are from the axis of symmetry. If the parabola's vertex is on the $$ x $$-axis, the equation has a single repeated root on the line of symmetry, and this distance term is zero, algebraically known as the **discriminant** $$ b^{2}-4ac=0 $$.

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