SECTIONA QUESTION 1 1. SIGNIFICANCE OF QUADRATIC FORMULA The quadratic formula defines the points ( \( x, 0 \) ) on the parabolic graph, where the parabola \( y^{\prime \prime}=a x^{2}+b x+c \) crosses the \( x \)-axis and it can be separated into two terms. The quadratics formula is: \( x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \) or \( x=\frac{-b}{2 a} \pm \frac{\sqrt{b^{2}-4 a c}}{2 a} \) 1.1. The first term \( \frac{-b}{2 a} \) describes the (i) \( \qquad \) , the second term \( \frac{\sqrt{b^{2}-4 a c}}{2 a} \) gives the (ii) \( \qquad \) if the parabola's vertex is on the \( x \) axis, then the corresponding equation has a single repeated root the line of symmetry and this distance term is zero, algebraically the (iii) \( \qquad \) . commonly known as \( b^{2}-4 a c=0 \). (3) 1.2. Solve for x using the quadratic formula (4)

Below is one way to answer the question parts: ────────────────────────────── 1.1. In the quadratic formula   x = (–b ± √(b² – 4ac))⁄(2a) we can view it as the sum of two parts:  • The first term, –b⁄(2a), represents (i) the x‐coordinate of the vertex of the parabola. In other words, it is the line (axis) of symmetry for the parabola.  • The second term, √(b² – 4ac)⁄(2a), gives (ii) the distance (offset) from the axis of symmetry to each x‑intercept. That is, it tells you how far the intersection points lie from the vertex along the x‑axis.  • When the vertex lies exactly on the x‑axis, the parabola touches the x‑axis at a single point. Algebraically, this occurs when the distance term is zero, meaning √(b² – 4ac) = 0. This condition is (iii) the discriminant being equal to zero (i.e. b²–4ac = 0). ────────────────────────────── 1.2. Solving for x using the quadratic formula To solve a quadratic equation of the form   ax² + bx + c = 0 follow these steps: 1. Write the equation in standard form. 2. Identify the coefficients: a, b, and c. 3. Compute the discriminant: D = b² – 4ac. 4. Substitute into the quadratic formula:   x = (–b ± √D)⁄(2a) For example, consider the quadratic equation:   x² – 5x + 6 = 0 Here: a = 1, b = –5, and c = 6. Step-by-step solution:  a. Compute the discriminant:   D = (–5)² – 4(1)(6) = 25 – 24 = 1  b. Substitute into the quadratic formula:   x = [–(–5) ± √1]⁄(2×1) = (5 ± 1)⁄2  c. This gives two solutions:   x = (5 + 1)⁄2 = 6⁄2 = 3   and  x = (5 – 1)⁄2 = 4⁄2 = 2 Thus, the solutions are x = 3 and x = 2. ────────────────────────────── This completes the answer for both parts 1.1 and 1.2.