1. **Problem Statement:** We are exploring the discriminant (Δ) of a quadratic equation and its implications on the roots and graph of the equation. 2. **Definition and Formula:** The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by the formula: $$\Delta = b^2 - 4ac$$ This value helps us determine the nature of the roots without solving the equation. 3. **Nature of Roots Based on Δ:** - If $\Delta > 0$, the equation has two distinct real roots. - If $\Delta = 0$, the equation has exactly one real root (a repeated root). - If $\Delta < 0$, the equation has two complex conjugate roots (no real roots). 4. **Case $\Delta = 0$ Example:** Consider $x^2 - 4x + 4 = 0$. Calculate $\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0$. This means the roots are equal: $x = \frac{-b}{2a} = \frac{4}{2} = 2$. 5. **Case $\Delta > 0$ Example:** Consider $x^2 - 5x + 6 = 0$. Calculate $\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0$. Roots are distinct real numbers: $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{5 \pm 1}{2}$$ So, $x=3$ or $x=2$. 6. **Case $\Delta < 0$ Explanation:** If $\Delta < 0$, roots are complex conjugates, meaning no real intersection with the x-axis. 7. **Effect of Changing $b$ on $\Delta$:** Since $\Delta = b^2 - 4ac$, increasing or decreasing $b$ changes $b^2$ and thus $\Delta$, affecting the root types. 8. **Equal Roots Condition for $2x^2 - 3x + c = 0$:** Set $\Delta = 0$: $$(-3)^2 - 4(2)(c) = 0 \Rightarrow 9 - 8c = 0 \Rightarrow c = \frac{9}{8} = 1.125$$ 9. **Perfect Square Discriminant:** If $\Delta$ is a perfect square, roots are rational and real. 10. **No Real Roots with Real Coefficients:** Yes, if $\Delta < 0$, roots are complex despite real coefficients. 11. **Real-life Application:** Discriminant helps in physics to determine if a projectile hits the ground (real roots) or not. 12. **Graph of $y = x^2 - 4x + 3$:** Calculate $\Delta = (-4)^2 - 4(1)(3) = 16 - 12 = 4 > 0$. Since $\Delta > 0$, the parabola cuts the x-axis at two points. 13. **Graph of $y = 2x^2 + 3x + 5$:** Calculate $\Delta = 3^2 - 4(2)(5) = 9 - 40 = -31 < 0$. Negative discriminant means the parabola does not intersect the x-axis and lies entirely above it (since $a=2>0$). Final answers are summarized above with examples and explanations.