You're right! If we are dealing with the inequality $Ax^2 + Bx + C \geq 0$, here's how we proceed:

Step-by-Step Solution

1. Determine the nature of the roots:

   • Calculate the discriminant: $\Delta = B^2 - 4AC$.
   • Based on $\Delta$:
     • If $\Delta > 0$: Two distinct real roots exist.
     • If $\Delta = 0$: One real root (a double root) exists.
     • If $\Delta < 0$: No real roots exist (parabola does not touch the $x$-axis).

2. Analyze the parabola's direction:

   • If $A > 0$, the parabola opens upwards.
   • If $A < 0$, the parabola opens downwards.

3. Critical points:

   • If $\Delta \geq 0$: Find the roots $x_1$ and $x_2$ using the quadratic formula: $x_1, x_2 = \frac{-B \pm \sqrt{\Delta}}{2A}.$

4. Sign of the inequality:

   • For $Ax^2 + Bx + C \geq 0$:
     • If the parabola opens upwards ($A > 0$):
       • The solution is $(-\infty, x_1] \cup [x_2, \infty)$ when $\Delta > 0$.
       • If $\Delta = 0$: The solution is $x = x_1$ (the double root).
       • If $\Delta < 0$: The solution is all real numbers ($(-\infty, \infty)$).
     • If the parabola opens downwards ($A < 0$):
       • The solution is $[x_1, x_2]$ when $\Delta > 0$.
       • If $\Delta = 0$: The solution is $x = x_1$.
       • If $\Delta < 0$: The solution is the empty set ($\emptyset$).

5. Boundary handling:

   • If the inequality is strict ($Ax^2 + Bx + C > 0$), exclude the roots from the solution (replace brackets with parentheses in the intervals).

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Would you like me to work through a specific example or provide a graph of this situation? Let me know!

5 Related Questions:

1. What happens to the inequality if $A$ or $C$ is zero?
2. Can you demonstrate graphically why the solution intervals depend on $\Delta$?
3. How would the solution change for $Ax^2 + Bx + C \leq 0$?
4. What role does completing the square play in solving this inequality?
5. How do inequalities involving $\geq$ differ from strict inequalities involving $>$?

Tip:

Always check the discriminant first to quickly determine the nature of the parabola and how it affects the inequality solution.