1. The problem: Understand how to solve the discriminant of a quadratic equation and use it to determine the nature of the roots. 2. The quadratic equation is generally written as $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants. 3. The discriminant $\Delta$ is given by the formula $$\Delta = b^2 - 4ac$$. 4. The discriminant helps us determine 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 $\Delta < 0$, there are two complex conjugate roots. 5. Example: Solve the discriminant for the quadratic equation $$2x^2 - 4x + 1 = 0$$. 6. Identify coefficients: $a=2$, $b=-4$, $c=1$. 7. Calculate the discriminant: $$\Delta = (-4)^2 - 4 \times 2 \times 1 = 16 - 8 = 8$$. 8. Since $\Delta = 8 > 0$, the equation has two distinct real roots. 9. To find the roots, use the quadratic formula: $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-4) \pm \sqrt{8}}{2 \times 2} = \frac{4 \pm 2\sqrt{2}}{4} = 1 \pm \frac{\sqrt{2}}{2}$$. 10. Final answer: The roots are $$x = 1 + \frac{\sqrt{2}}{2}$$ and $$x = 1 - \frac{\sqrt{2}}{2}$$.