Quadratic Equations 82B120

1. **Solve the quadratic equation** $m^2 - 5m - 14 = 0$. 2. The standard form of a quadratic equation is $$ax^2 + bx + c = 0$$. 3. We can solve by factoring, completing the square, or using the quadratic formula: $$m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. For the first equation, $a=1$, $b=-5$, $c=-14$. 5. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times (-14) = 25 + 56 = 81$$ 6. Since $\Delta > 0$, two real roots exist. 7. Calculate the roots: $$m = \frac{-(-5) \pm \sqrt{81}}{2 \times 1} = \frac{5 \pm 9}{2}$$ 8. Roots: $$m_1 = \frac{5 + 9}{2} = \frac{14}{2} = 7$$ $$m_2 = \frac{5 - 9}{2} = \frac{-4}{2} = -2$$ --- 9. **Solve the quadratic equation** $2m^2 + 2m - 12 = 0$. 10. Here, $a=2$, $b=2$, $c=-12$. 11. Calculate the discriminant: $$\Delta = 2^2 - 4 \times 2 \times (-12) = 4 + 96 = 100$$ 12. Two real roots exist. 13. Calculate the roots: $$m = \frac{-2 \pm \sqrt{100}}{2 \times 2} = \frac{-2 \pm 10}{4}$$ 14. Roots: $$m_1 = \frac{-2 + 10}{4} = \frac{8}{4} = 2$$ $$m_2 = \frac{-2 - 10}{4} = \frac{-12}{4} = -3$$ --- 15. **Solve the quadratic equation** $x^2 + 5x + 6 = 0$. 16. Here, $a=1$, $b=5$, $c=6$. 17. Calculate the discriminant: $$\Delta = 5^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$ 18. Two real roots exist. 19. Calculate the roots: $$x = \frac{-5 \pm \sqrt{1}}{2 \times 1} = \frac{-5 \pm 1}{2}$$ 20. Roots: $$x_1 = \frac{-5 + 1}{2} = \frac{-4}{2} = -2$$ $$x_2 = \frac{-5 - 1}{2} = \frac{-6}{2} = -3$$ **Final answers:** - For $m^2 - 5m - 14 = 0$: $m=7$ or $m=-2$ - For $2m^2 + 2m - 12 = 0$: $m=2$ or $m=-3$ - For $x^2 + 5x + 6 = 0$: $x=-2$ or $x=-3$