1. The problem: Understand the discriminant rules for quadratic equations. 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. Important rules about the discriminant: - If $\Delta > 0$, the quadratic equation has two distinct real roots. - If $\Delta = 0$, the quadratic equation has exactly one real root (a repeated root). - If $\Delta < 0$, the quadratic equation has two complex conjugate roots (no real roots). 5. These rules help us determine the nature of the roots without solving the equation. 6. Example: For $x^2 - 4x + 3 = 0$, $a=1$, $b=-4$, $c=3$. Calculate $$\Delta = (-4)^2 - 4 \times 1 \times 3 = 16 - 12 = 4 > 0$$. So, two distinct real roots exist. 7. Summary: Use $$\Delta = b^2 - 4ac$$ to find the discriminant and apply the rules above to know the roots' nature.