1. **State the problem:** We want to find values of $a$ such that the quadratic equation $$x^2 - (a^2 - 2)x - a^2 + 3a + 2 = 0$$ has one root greater than 1 and the other root less than 1. 2. **Recall the quadratic formula and root properties:** For a quadratic equation $x^2 + bx + c = 0$, the sum of roots is $-b$ and the product of roots is $c$. 3. **Identify sum and product of roots:** Here, the equation is $$x^2 - (a^2 - 2)x - a^2 + 3a + 2 = 0,$$ so - Sum of roots $S = a^2 - 2$ - Product of roots $P = -a^2 + 3a + 2$ 4. **Condition for roots around 1:** Let the roots be $r_1$ and $r_2$. We want one root $>1$ and the other $<1$. This means the value of the quadratic at $x=1$ must be negative (since the parabola opens upwards and the roots lie on either side of 1): $$f(1) = 1^2 - (a^2 - 2)(1) - a^2 + 3a + 2 < 0$$ Simplify: $$1 - a^2 + 2 - a^2 + 3a + 2 < 0$$ $$5 + 3a - 2a^2 < 0$$ 5. **Solve inequality:** $$-2a^2 + 3a + 5 < 0$$ Multiply both sides by $-1$ (reverse inequality): $$2a^2 - 3a - 5 > 0$$ 6. **Find roots of quadratic $2a^2 - 3a - 5 = 0$:** $$a = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} = \frac{3 \pm \sqrt{9 + 40}}{4} = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4}$$ So roots are: - $$a = \frac{3 + 7}{4} = \frac{10}{4} = 2.5$$ - $$a = \frac{3 - 7}{4} = \frac{-4}{4} = -1$$ 7. **Determine intervals where inequality holds:** Since the parabola opens upwards, $2a^2 - 3a - 5 > 0$ for $a < -1$ or $a > 2.5$. 8. **Check discriminant for real roots:** Discriminant of original quadratic: $$\Delta = (a^2 - 2)^2 - 4 \cdot 1 \cdot (-a^2 + 3a + 2) = (a^2 - 2)^2 + 4a^2 - 12a - 8$$ Simplify: $$a^4 - 4a^2 + 4 + 4a^2 - 12a - 8 = a^4 - 12a - 4$$ For real roots, $$\Delta \geq 0$$. 9. **Summary:** - Roots are real if $$a^4 - 12a - 4 \geq 0$$ - One root greater than 1 and the other less than 1 if $$a < -1$$ or $$a > 2.5$$ **Final answer:** The values of $a$ for which one root is greater than 1 and the other less than 1 are those satisfying $$a < -1$$ or $$a > 2.5$$, provided the discriminant $$a^4 - 12a - 4 \geq 0$$ to ensure real roots.