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 away from the axis of symmetry. repeated root on the line of symmetry, and this distance term is zero, algebraically, the (iii) commonly known as \( b^{2}-4 a c=0 \). GRAND TOTAL: 100 MARKS

Did you know that the quadratic formula has roots that stretch back to ancient civilizations? The Babylonians were using methods akin to the quadratic formula as far back as 2000 BCE! Although they did not have a formalized formula, they solved quadratic equations through geometric approaches, showcasing their impressive understanding of mathematics long before we simplified it into a neat equation that we use today. In the real world, the quadratic formula is not just for academics—it's a powerful tool in various fields! Engineers use it to determine projectile motion, like the path of a thrown ball, while economists apply it to maximize profit or minimize costs by analyzing revenue or cost functions. The versatility of the quadratic formula makes it relevant not just in textbooks, but in everyday problem-solving scenarios, from building bridges to budgeting!