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 \( b^{2}-4 a c=0 \).

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** \( b^{2}-4 a c=0 \).