1. The problem is to understand why the quadratic formula (qfs) might be considered incorrect or to verify its correctness. 2. The quadratic formula is used to solve quadratic equations of the form $ax^2 + bx + c = 0$ and is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. Important rules: - The discriminant $\Delta = b^2 - 4ac$ determines the nature of the roots. - If $\Delta > 0$, there are two distinct real roots. - If $\Delta = 0$, there is one real root (a repeated root). - If $\Delta < 0$, there are two complex roots. 4. To verify the formula, start from the quadratic equation: $$ax^2 + bx + c = 0$$ Divide both sides by $a$ (assuming $a \neq 0$): $$\cancel{a}x^2 + \frac{b}{\cancel{a}}x + \frac{c}{a} = 0 \implies x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$ 5. Complete the square: $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$ Add $\left(\frac{b}{2a}\right)^2$ to both sides: $$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$ 6. This gives: $$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{c}{a} = \frac{b^2 - 4ac}{4a^2}$$ 7. Taking the square root of both sides: $$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$ 8. Finally, solve for $x$: $$x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 9. This confirms the quadratic formula is correct. If you believe the quadratic formula is incorrect, please check the values of $a$, $b$, and $c$ or the discriminant calculation.