1. **State the problems:** Solve the quadratic equations from 29 to 36. 2. **Recall the quadratic formula:** For any quadratic equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Solve each equation step-by-step:** **29.** $m^2 + 3m + 2 = 0$ - Factor: $(m + 1)(m + 2) = 0$ - Solutions: $m = -1, -2$ **30.** $n^2 - 9n + 18 = 0$ - Factor: $(n - 3)(n - 6) = 0$ - Solutions: $n = 3, 6$ **31.** $x^2 + 5x - 14 = 0$ - Factor: $(x + 7)(x - 2) = 0$ - Solutions: $x = -7, 2$ **32.** $y^2 + 11v - 26 = 0$ - Assuming typo, replace $v$ with $y$: $y^2 + 11y - 26 = 0$ - Use quadratic formula: $$y = \frac{-11 \pm \sqrt{11^2 - 4 \times 1 \times (-26)}}{2} = \frac{-11 \pm \sqrt{121 + 104}}{2} = \frac{-11 \pm \sqrt{225}}{2}$$ - Simplify: $$y = \frac{-11 \pm 15}{2}$$ - Solutions: $$y = \frac{-11 + 15}{2} = 2, \quad y = \frac{-11 - 15}{2} = -13$$ **33.** $t^2 + 15t = -36$ - Rewrite: $t^2 + 15t + 36 = 0$ - Factor: $(t + 3)(t + 12) = 0$ - Solutions: $t = -3, -12$ **34.** $n^2 - 5n = 24$ - Rewrite: $n^2 - 5n - 24 = 0$ - Factor: $(n - 8)(n + 3) = 0$ - Solutions: $n = 8, -3$ **35.** $a^2 + 5a - 20 = 30$ - Rewrite: $a^2 + 5a - 50 = 0$ - Use quadratic formula: $$a = \frac{-5 \pm \sqrt{5^2 - 4 \times 1 \times (-50)}}{2} = \frac{-5 \pm \sqrt{25 + 200}}{2} = \frac{-5 \pm \sqrt{225}}{2}$$ - Simplify: $$a = \frac{-5 \pm 15}{2}$$ - Solutions: $$a = \frac{-5 + 15}{2} = 5, \quad a = \frac{-5 - 15}{2} = -10$$ **36.** $y^2 - 2y - 8 = 7$ - Rewrite: $y^2 - 2y - 15 = 0$ - Factor: $(y - 5)(y + 3) = 0$ - Solutions: $y = 5, -3$