Grade 11 Mathematics/Investigation 8 Term 12025 NSC c) Isolate the \( x \) on the left by moving the other term to the right \( \qquad \) d) Write the right-hand side as a single fraction \( \qquad \) If you have followed the steps correctly you have established quadratic formula, i.e. for a quadratic equation of the form \( a x^{2}+b x+c=0 \), solutions are: \( 2.7 x= \) (2) \( \qquad \) Further questions to ponder: 3. Who was the first person to establish this formula and when? (2) 4. Are there other ways to establish the formula? (5) 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 \( b^{2}-4 a c=0 \). GRAND TOTAL: 100 MARKS Copyright reserved Please turn over