1. Let's start by stating the problem: Understanding the discriminant in quadratic equations and how it helps determine the nature of the roots. 2. A quadratic equation is generally written as $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants and $a \neq 0$. 3. The discriminant $\Delta$ is given by the formula $$\Delta = b^2 - 4ac$$. 4. The discriminant tells us about the nature of the roots of the quadratic equation: - 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 (no real roots). 5. To see why, recall the quadratic formula for roots: $$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$ 6. The term under the square root, $\sqrt{\Delta}$, determines whether the roots are real or complex: - If $\Delta$ is positive, $\sqrt{\Delta}$ is a real number, so roots are real and different. - If $\Delta$ is zero, $\sqrt{\Delta} = 0$, so both roots are the same. - If $\Delta$ is negative, $\sqrt{\Delta}$ is imaginary, so roots are complex. 7. Example: For the equation $2x^2 + 3x - 2 = 0$, calculate the discriminant: $$\Delta = 3^2 - 4 \times 2 \times (-2) = 9 + 16 = 25$$ Since $\Delta = 25 > 0$, there are two distinct real roots. 8. Using the quadratic formula: $$x = \frac{-3 \pm \sqrt{25}}{2 \times 2} = \frac{-3 \pm 5}{4}$$ 9. Calculate each root: $$x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}$$ $$x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$$ 10. So the roots are $x = \frac{1}{2}$ and $x = -2$. This shows how the discriminant helps us understand the roots of a quadratic equation clearly and efficiently.