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)

Question

UpStudy AI · Accepted Answer

**c)** Starting with  
$$
x+\frac{b}{2a}=\pm \sqrt{\frac{b^2-4ac}{4a^2}},
$$  
subtract $$\frac{b}{2a}$$ from both sides to isolate $$x$$:  
$$
x=-\frac{b}{2a}\pm\sqrt{\frac{b^2-4ac}{4a^2}}.
$$

**d)** The square root term can be written as a fraction with denominator $$2a$$ since  
$$
\sqrt{\frac{b^2-4ac}{4a^2}}=\frac{\sqrt{b^2-4ac}}{2a}.
$$  
Thus,  
$$
x=-\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac}}{2a}.
$$  
Writing the right side as a single fraction gives the quadratic formula:  
$$
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.
$$

**3.** The earliest known derivation of a method equivalent to the quadratic formula is found in the work of the ancient Babylonians around $$2000$$ BC. However, the systematic algebraic solution of quadratic equations—often associated with the method of "completing the square"—was established by the Persian mathematician $$\textbf{Al-Khwarizmi}$$ around $$825$$ AD.

**4.** Yes, there are other methods to establish the quadratic formula. Some alternative approaches include:  
- **Completing the square:** As demonstrated above, this classical method transforms the quadratic equation into a perfect square before taking square roots.  
- **Using Vieta's formulas:** By expressing the sum and product of the roots in terms of the coefficients, one can derive relationships that lead to the quadratic formula.  
- **Geometric methods:** Some derivations use geometric constructions to interpret and solve the quadratic equation.  
- **Method of undetermined coefficients or transformation techniques:** These algebraic methods simplify the quadratic equation to a depressed form before solving.

Each method, while conceptually different, ultimately leads to the same quadratic formula.
**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.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions