Quadratic Inequality Ebe5F9

1. **State the problem:** Solve the quadratic inequality $ax^2 + bx + c > 0$ (or $< 0$, $\geq 0$, $\leq 0$) where $a$, $b$, and $c$ are constants. 2. **Formula and rules:** The solution depends on the roots of the quadratic equation $ax^2 + bx + c = 0$. 3. **Find the roots:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Analyze the discriminant $\Delta = b^2 - 4ac$:** - If $\Delta > 0$, there are two distinct real roots. - If $\Delta = 0$, there is one real root (a repeated root). - If $\Delta < 0$, there are no real roots. 5. **Determine intervals:** The parabola opens upward if $a > 0$ and downward if $a < 0$. 6. **Test intervals between and outside roots:** - For $a > 0$ and $ax^2 + bx + c > 0$, the solution is $x < x_1$ or $x > x_2$ where $x_1$ and $x_2$ are roots. - For $a > 0$ and $ax^2 + bx + c < 0$, the solution is $x_1 < x < x_2$. 7. **Example:** Solve $x^2 - 5x + 6 > 0$. - Find roots: $$x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}$$ - Roots are $x_1 = 2$, $x_2 = 3$. - Since $a=1 > 0$ and inequality is $> 0$, solution is $x < 2$ or $x > 3$. **Final answer:** $x \in (-\infty, 2) \cup (3, \infty)$