Question

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)

Ask by Smith Craig. in South Africa
Mar 16,2025

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**c)** \[ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \] **d)** \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] **3.** The quadratic formula was first established by the Persian mathematician **Al-Khwarizmi** around \(825\) AD. **4.** Yes, other methods to derive the quadratic formula include completing the square, using Vieta's formulas, geometric constructions, and algebraic techniques like the method of undetermined coefficients.

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The Deep Dive

Did you know the quadratic formula, \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is often attributed to ancient civilizations? While the formula's recognizable form emerged in Europe during the 17th century, the Babylonians were solving quadratic equations as far back as 2000 BC! Their methods may not have looked like our modern algebra, but their understanding of these problems showed the world how fascinating mathematics can be. As for alternative ways to establish the quadratic formula, one method is completing the square! This technique involves rearranging the quadratic equation into a perfect square trinomial, which makes it easier to solve for \(x\). Another approach is using graphical methods, where you can visualize the parabolic shape of a quadratic equation and find the moments where it intersects the x-axis. Both methods add a unique flavor to developing solutions!

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