Further questions to ponder: 3. Who was the first person to establish this formula and when? 4. Are there other ways to establish the formula? SECTION C: SIGNIFICANCE OF THE QUADRATIC FORMULA 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) \( \frac{\sqrt{b^{2}-4 a c}}{2 a} \), gives the (ii) If the parabola's vertex is on the \( x-a x i s \), then the corresponding equation has a single \( x=-\frac{b}{2 a} \). The second term repeated root on the line of symmetry, and this distance term is zero, algebraically, the (iii) the are away from the axis of symmetry. commonly known as \( b^{2}-4 a c=0 \). GRAND TOTAL: 100 MARKS ( 3 ,

**Further Questions to Ponder:** 3. The first person to establish the quadratic formula was **Al-Khwarizmi**, a Persian mathematician, around the 9th century. 4. Yes, there are multiple ways to derive the quadratic formula, including factoring, graphical methods, and using symmetry. **SECTION C: SIGNIFICANCE OF THE QUADRATIC FORMULA** 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. It can be separated into two terms: \[ x = \frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \] \[ x = -\frac{b}{2 a} \pm \frac{\sqrt{b^{2}-4 a c}}{2 a} \] 1. The first term \( -\frac{b}{2 a} \) describes the **axis of symmetry**. 2. \( \frac{\sqrt{b^{2}-4 a c}}{2 a} \) gives the **distance from the axis of symmetry to each root**. 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, the **discriminant equals zero**, commonly known as \( b^{2}-4 a c = 0 \).