1. Let's start with Quadratic Equations and their roots and discriminant. A quadratic equation is of the form $$ax^2 + bx + c = 0$$ where $a \neq 0$. 2. The roots of the quadratic equation can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. The expression under the square root, $$\Delta = b^2 - 4ac$$, is called the discriminant. 4. The discriminant tells us about the nature of the roots: - If $$\Delta > 0$$, there are two distinct real roots. - If $$\Delta = 0$$, there is exactly one real root (a repeated root). - If $$\Delta < 0$$, the roots are complex (no real roots). 5. Example: Solve $$2x^2 - 4x - 6 = 0$$. 6. Calculate the discriminant: $$\Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$ 7. Since $$\Delta = 64 > 0$$, there are two distinct real roots. 8. Use the quadratic formula: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ 9. Calculate each root: $$x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3$$ $$x_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1$$ 10. So, the roots are $$x = 3$$ and $$x = -1$$. This completes the explanation of quadratic equations, roots, and discriminant.