1. **State the problem:** We have the quadratic equation $x^2 - (m - 1)x + m + 2 = 0$ and want to find the values of $m$ for different root conditions. 2. **Recall discriminant formula:** The discriminant $\Delta$ determines the nature of roots. $$\Delta = b^2 - 4ac$$ Here, $a=1$, $b=-(m-1)$, and $c=m+2$. 3. **Calculate the discriminant:** $$\Delta = (-(m-1))^2 - 4(1)(m+2) = (m-1)^2 - 4(m+2)$$ Simplify: $$= m^2 - 2m + 1 - 4m - 8 = m^2 - 6m - 7$$ 4. **Case 1: No real roots** means: $$\Delta < 0 \implies m^2 - 6m - 7 < 0$$ Solve the inequality by factoring the quadratic expression: $$m^2 - 6m - 7 = (m - 7)(m + 1)$$ Check the inequality: Between roots the expression is negative: $$-1 < m < 7$$ 5. **Case 2: Two distinct real roots** means: $$\Delta > 0 \implies m^2 - 6m - 7 > 0$$ This holds outside the roots of the parabola: $$m < -1 \text{ or } m > 7$$ 6. **Case 3: Two distinct roots of opposite signs** means: - The product of roots must be negative. - From quadratic formula: Product of roots = $c/a = m+2$ So: $$m + 2 < 0 \implies m < -2$$ Also from case 2, roots must exist (discriminant $>0$). Check which $m$ satisfy both: $$m < -2 \\ \text{and} \\ (m < -1 \text{ or } m > 7)$$ So it is: $$m < -2$$ 7. **Case 4: Two strictly positive distinct roots** means: - Roots are distinct: $\Delta > 0$ - Both roots positive: sum and product of roots are positive. Sum of roots: $\frac{-b}{a} = m-1$ Product of roots: $\frac{c}{a} = m + 2$ Conditions: $$m - 1 > 0 \implies m > 1$$ $$m + 2 > 0 \implies m > -2$$ Together with $\Delta > 0 \implies m < -1 \text{ or } m > 7$ Combine all: $$m > 7$$ 8. **Case 5: Two strictly negative distinct roots** means: - Roots are distinct: $\Delta > 0$ - Both roots negative: sum and product of roots are positive and negative respectively: Sum of roots $= m - 1 < 0 \implies m < 1$ Product of roots $= m + 2 > 0 \implies m > -2$ With $\Delta > 0$, which is $m < -1$ or $m > 7$. Combine all: $$-2 < m < -1$$ 9. **Case 6: Double root** means: $$\Delta = 0 \implies m^2 - 6m - 7 = 0$$ Solve: $$m = \frac{6 \pm \sqrt{36 + 28}}{2} = \frac{6 \pm \sqrt{64}}{2} = \frac{6 \pm 8}{2}$$ Possible values: $$m = 7 \quad \text{or} \quad m = -1$$ **Summary:** 1) No real roots: $-1 < m < 7$ 2) Two distinct real roots: $m < -1$ or $m > 7$ 3) Two distinct roots of opposite signs: $m < -2$ 4) Two strictly positive distinct roots: $m > 7$ 5) Two strictly negative distinct roots: $-2 < m < -1$ 6) Double root: $m = -1$ or $m = 7$