1. **Problem 1:** Solve $8x^2 - 2x - 15 = 0$ by factorization. - Multiply $a \times c = 8 \times (-15) = -120$. - Find two numbers that multiply to $-120$ and add to $-2$: these are $10$ and $-12$. - Rewrite middle term: $8x^2 + 10x - 12x - 15 = 0$. - Factor by grouping: $2x(4x + 5) - 3(4x + 5) = 0$. - Factor out common binomial: $(2x - 3)(4x + 5) = 0$. - Set each factor to zero: $2x - 3 = 0 \Rightarrow x = \frac{3}{2}$, $4x + 5 = 0 \Rightarrow x = -\frac{5}{4}$. 2. **Problem 2:** Solve $x^2 + 10x + 24 = 0$ by factorization. - Find two numbers that multiply to $24$ and add to $10$: $6$ and $4$. - Factor: $(x + 6)(x + 4) = 0$. - Solutions: $x = -6$, $x = -4$. 3. **Problem 3:** Solve $9x^2 + 6x - 8 = 0$ by factorization. - Multiply $a \times c = 9 \times (-8) = -72$. - Find two numbers that multiply to $-72$ and add to $6$: $12$ and $-6$. - Rewrite middle term: $9x^2 + 12x - 6x - 8 = 0$. - Factor by grouping: $3x(3x + 4) - 2(3x + 4) = 0$. - Factor out common binomial: $(3x - 2)(3x + 4) = 0$. - Solutions: $x = \frac{2}{3}$, $x = -\frac{4}{3}$. 4. **Problem 4:** Solve $x^2 - 8x + 15 = 0$ by factorization. - Find two numbers that multiply to $15$ and add to $-8$: $-3$ and $-5$. - Factor: $(x - 3)(x - 5) = 0$. - Solutions: $x = 3$, $x = 5$. 5. **Problem 5:** Solve $2x^2 - 16x + 30 = 0$ by factorization. - Divide entire equation by 2: $x^2 - 8x + 15 = 0$. - Factor as in problem 4: $(x - 3)(x - 5) = 0$. - Solutions: $x = 3$, $x = 5$. 6. **Problem 6:** Solve $x^2 + 9x + 20 = 0$ by factorization. - Find two numbers that multiply to $20$ and add to $9$: $5$ and $4$. - Factor: $(x + 5)(x + 4) = 0$. - Solutions: $x = -5$, $x = -4$. 7. **Problem 7:** Solve $10x^2 - 40x + 40 = 0$ by factorization. - Divide entire equation by 10: $x^2 - 4x + 4 = 0$. - Recognize perfect square: $(x - 2)^2 = 0$. - Solution: $x = 2$. 8. **Problem 8:** Solve $36x^2 - 84x + 24 = 0$ by factorization. - Divide entire equation by 12: $3x^2 - 7x + 2 = 0$. - Multiply $a \times c = 3 \times 2 = 6$. - Find two numbers that multiply to $6$ and add to $-7$: $-6$ and $-1$. - Rewrite middle term: $3x^2 - 6x - x + 2 = 0$. - Factor by grouping: $3x(x - 2) - 1(x - 2) = 0$. - Factor out common binomial: $(3x - 1)(x - 2) = 0$. - Solutions: $x = \frac{1}{3}$, $x = 2$. **Final answers:** 1. $x = \frac{3}{2}, -\frac{5}{4}$ 2. $x = -6, -4$ 3. $x = \frac{2}{3}, -\frac{4}{3}$ 4. $x = 3, 5$ 5. $x = 3, 5$ 6. $x = -5, -4$ 7. $x = 2$ 8. $x = \frac{1}{3}, 2$