Quadratic Inequality F919Cf

1. **State the problem:** Solve the inequality $$x^2 - 7x + 12 > 0$$. 2. **Recall the formula and rules:** This is a quadratic inequality. To solve it, first find the roots of the quadratic equation $$x^2 - 7x + 12 = 0$$ by factoring or using the quadratic formula. 3. **Factor the quadratic:** $$x^2 - 7x + 12 = (x - 3)(x - 4)$$. 4. **Find the roots:** Set each factor equal to zero: $$x - 3 = 0 \Rightarrow x = 3$$ $$x - 4 = 0 \Rightarrow x = 4$$ 5. **Analyze the inequality:** The quadratic opens upwards (coefficient of $$x^2$$ is positive), so the parabola is above the x-axis outside the roots and below between the roots. 6. **Write the solution:** $$x^2 - 7x + 12 > 0$$ when $$x < 3$$ or $$x > 4$$. 7. **Final answer:** $$(-\infty, 3) \cup (4, \infty)$$.