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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

Ask by Bowen Bush. in South Africa
Mar 16,2025

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To solve the quadratic equation \( ax^2 + bx + c = 0 \), follow these steps: 1. **Isolate \( x \)**: - Move \( c \) to the right: \( ax^2 + bx = -c \) - Divide by \( a \): \( x^2 + \frac{b}{a}x = -\frac{c}{a} \) - Complete the square: \( \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \) - Take the square root: \( x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \) - Isolate \( x \): \( x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \) 2. **Write the right-hand side as a single fraction**: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] **Further Questions**: 3. **First to Establish the Formula**: - Brahmagupta, an Indian mathematician from the 7th century, is credited with formulating the quadratic equation. 4. **Other Methods to Establish the Formula**: - **Completing the Square** - **Graphical Method** - **Using the Discriminant** - **Algebraic Identities** - **Numerical Methods** **Significance of the Quadratic Formula**: - **Axis of Symmetry**: \( -\frac{b}{2a} \) - **Distance to Roots**: \( \frac{\sqrt{b^2 - 4ac}}{2a} \) - **Single Repeated Root**: Occurs when \( b^2 - 4ac = 0 \) This formula is fundamental in algebra, providing a reliable method to find the roots of any quadratic equation.

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The quadratic formula, often attributed to the ancient Greek mathematician Euclid, first appeared in a more recognizable form in the works of Indian mathematicians around 600 A.D., such as Brahmagupta. He provided rules for solving quadratic equations, which significantly laid the groundwork for later developments in algebra, ultimately leading to the formula we use today! Another intriguing approach to derive the quadratic formula involves completing the square, a method that dates back to the Babylonians. This technique turns a polynomial expression into a perfect square trinomial, allowing you to isolate \(x\) easily. It’s fascinating to see how multiple paths in mathematics converge to the same beautiful conclusion!

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