1. **Problem:** Solve for $x$ in the equation $x - 5 = -8$ and then evaluate $(x^2 - 5)(x^3 - x)$. 2. **Step 1: Solve for $x$.** $$x - 5 = -8$$ Add 5 to both sides: $$x - 5 + 5 = -8 + 5$$ $$x = -3$$ 3. **Step 2: Evaluate $(x^2 - 5)(x^3 - x)$ at $x = -3$.** Calculate each part: $$x^2 - 5 = (-3)^2 - 5 = 9 - 5 = 4$$ $$x^3 - x = (-3)^3 - (-3) = -27 + 3 = -24$$ 4. **Step 3: Multiply the results:** $$(x^2 - 5)(x^3 - x) = 4 imes (-24) = -96$$ --- 5. **Problem:** Factor the quadratic expressions. 6. **Step 1: Factor $x^2 + 6x - 16$.** Find two numbers that multiply to $-16$ and add to $6$: $8$ and $-2$. $$x^2 + 6x - 16 = (x + 8)(x - 2)$$ 7. **Step 2: Factor $x^2 - 6x + 9$.** Find two numbers that multiply to $9$ and add to $-6$: $-3$ and $-3$. $$x^2 - 6x + 9 = (x - 3)^2$$ 8. **Step 3: Factor $x^2 - 6x - 27$.** Find two numbers that multiply to $-27$ and add to $-6$: $-9$ and $3$. $$x^2 - 6x - 27 = (x - 9)(x + 3)$$ 9. **Step 4: Factor $p^2 - p - 20$.** Find two numbers that multiply to $-20$ and add to $-1$: $-5$ and $4$. $$p^2 - p - 20 = (p - 5)(p + 4)$$ 10. **Step 5: Factor $x^2 - 2x - 15$.** Find two numbers that multiply to $-15$ and add to $-2$: $-5$ and $3$. $$x^2 - 2x - 15 = (x - 5)(x + 3)$$ 11. **Step 6: Factor $p^2 - 4p - 21$.** Find two numbers that multiply to $-21$ and add to $-4$: $-7$ and $3$. $$p^2 - 4p - 21 = (p - 7)(p + 3)$$ 12. **Step 7: Factor $p^2 + p - 20$.** Find two numbers that multiply to $-20$ and add to $1$: $5$ and $-4$. $$p^2 + p - 20 = (p + 5)(p - 4)$$ 13. **Step 8: Factor $k^2 - 3k - 40$.** Find two numbers that multiply to $-40$ and add to $-3$: $-8$ and $5$. $$k^2 - 3k - 40 = (k - 8)(k + 5)$$ 14. **Step 9: Factor $m^2 + 9m + 20$.** Find two numbers that multiply to $20$ and add to $9$: $5$ and $4$. $$m^2 + 9m + 20 = (m + 5)(m + 4)$$ --- 15. **Problem:** Rearrange and factor the following expressions. 16. **Step 1: $x^2 + 33 + 14x$** Rearranged: $x^2 + 14x + 33$ Find two numbers that multiply to $33$ and add to $14$: $11$ and $3$. $$(x + 11)(x + 3)$$ 17. **Step 2: $-13p + p^2 + 36$** Rearranged: $p^2 - 13p + 36$ Find two numbers that multiply to $36$ and add to $-13$: $-9$ and $-4$. $$(p - 9)(p - 4)$$ 18. **Step 3: $-30 + m^2 - m$** Rearranged: $m^2 - m - 30$ Find two numbers that multiply to $-30$ and add to $-1$: $-6$ and $5$. $$(m - 6)(m + 5)$$ 19. **Step 4: $11n + n^2 + 18$** Rearranged: $n^2 + 11n + 18$ Find two numbers that multiply to $18$ and add to $11$: $9$ and $2$. $$(n + 9)(n + 2)$$ 20. **Step 5: $x^2 + 27 + 12x$** Rearranged: $x^2 + 12x + 27$ Find two numbers that multiply to $27$ and add to $12$: $9$ and $3$. $$(x + 9)(x + 3)$$ 21. **Step 6: $x^2 + 90 - 19x$** Rearranged: $x^2 - 19x + 90$ Find two numbers that multiply to $90$ and add to $-19$: $-10$ and $-9$. $$(x - 10)(x - 9)$$ 22. **Step 7: $x^2 + x - 132$** Find two numbers that multiply to $-132$ and add to $1$: $12$ and $-11$. $$(x + 12)(x - 11)$$ 23. **Step 8: $a^2 + 90 - 47a$** Rearranged: $a^2 - 47a + 90$ Find two numbers that multiply to $90$ and add to $-47$: $-45$ and $-2$. $$(a - 45)(a - 2)$$ 24. **Step 9: $10m + m^2 + 16$** Rearranged: $m^2 + 10m + 16$ Find two numbers that multiply to $16$ and add to $10$: $8$ and $2$. $$(m + 8)(m + 2)$$ --- 25. **Problem:** Solve systems of equations using substitution. 26. **System 22:** $$\begin{cases} 3x + y = 9 \\ x - 4y = -10 \end{cases}$$ 27. **Step 1: Solve first equation for $y$: ** $$y = 9 - 3x$$ 28. **Step 2: Substitute into second equation:** $$x - 4(9 - 3x) = -10$$ $$x - 36 + 12x = -10$$ $$13x - 36 = -10$$ $$13x = 26$$ $$x = 2$$ 29. **Step 3: Find $y$:** $$y = 9 - 3(2) = 9 - 6 = 3$$ 30. **Solution:** $(x, y) = (2, 3)$ 31. **System 23:** $$\begin{cases} 2x + 5y = 7 \\ x + 3y = 4 \end{cases}$$ 32. **Step 1: Solve second equation for $x$: ** $$x = 4 - 3y$$ 33. **Step 2: Substitute into first equation:** $$2(4 - 3y) + 5y = 7$$ $$8 - 6y + 5y = 7$$ $$8 - y = 7$$ $$-y = -1$$ $$y = 1$$ 34. **Step 3: Find $x$:** $$x = 4 - 3(1) = 4 - 3 = 1$$ 35. **Solution:** $(x, y) = (1, 1)$ --- 36. **Problem:** Solve systems of equations using elimination. 37. **System 24:** $$\begin{cases} 3x + 4y = -7 \\ 3x - 3y = 21 \end{cases}$$ 38. **Step 1: Subtract second equation from first:** $$(3x + 4y) - (3x - 3y) = -7 - 21$$ $$3x + 4y - 3x + 3y = -28$$ $$7y = -28$$ $$y = -4$$ 39. **Step 2: Substitute $y$ into first equation:** $$3x + 4(-4) = -7$$ $$3x - 16 = -7$$ $$3x = 9$$ $$x = 3$$ 40. **Solution:** $(x, y) = (3, -4)$ 41. **System 25:** $$\begin{cases} 2x - 2y = -2 \\ 4x - 5y = -9 \end{cases}$$ 42. **Step 1: Multiply first equation by 2:** $$4x - 4y = -4$$ 43. **Step 2: Subtract second equation from this:** $$(4x - 4y) - (4x - 5y) = -4 - (-9)$$ $$4x - 4y - 4x + 5y = 5$$ $$y = 5$$ 44. **Step 3: Substitute $y$ into first equation:** $$2x - 2(5) = -2$$ $$2x - 10 = -2$$ $$2x = 8$$ $$x = 4$$ 45. **Solution:** $(x, y) = (4, 5)$ --- 46. **Problem:** Simplify radical expressions. 47. **Step 1: Simplify $7\sqrt{20} - 5\sqrt{32}$.** $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ $$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$$ $$7\sqrt{20} - 5\sqrt{32} = 7(2\sqrt{5}) - 5(4\sqrt{2}) = 14\sqrt{5} - 20\sqrt{2}$$ 48. **Step 2: Simplify $2\sqrt{18} - 5\sqrt{8} + 4\sqrt{50}$.** $$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$ $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$ $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$ $$2\sqrt{18} - 5\sqrt{8} + 4\sqrt{50} = 2(3\sqrt{2}) - 5(2\sqrt{2}) + 4(5\sqrt{2}) = 6\sqrt{2} - 10\sqrt{2} + 20\sqrt{2} = (6 - 10 + 20)\sqrt{2} = 16\sqrt{2}$$ --- 49. **Problem:** Simplify algebraic fractions. 50. **Step 1: Simplify $\frac{1 + \frac{1}{y}}{\frac{1}{y}}$.** Rewrite numerator: $$1 + \frac{1}{y} = \frac{y}{y} + \frac{1}{y} = \frac{y + 1}{y}$$ Expression becomes: $$\frac{\frac{y + 1}{y}}{\frac{1}{y}} = \frac{y + 1}{y} \times \frac{y}{1} = y + 1$$ 51. **Step 2: Simplify $\frac{\frac{a}{b} - 4}{\frac{x}{b} - b}$.** Rewrite denominator: $$\frac{x}{b} - b = \frac{x}{b} - \frac{b^2}{b} = \frac{x - b^2}{b}$$ Expression becomes: $$\frac{\frac{a}{b} - 4}{\frac{x - b^2}{b}} = \frac{\frac{a - 4b}{b}}{\frac{x - b^2}{b}} = \frac{a - 4b}{b} \times \frac{b}{x - b^2} = \frac{a - 4b}{x - b^2}$$ 52. **Step 3: Simplify $\frac{\frac{a}{x} - a}{x + \frac{y}{x}}$.** Rewrite numerator: $$\frac{a}{x} - a = \frac{a}{x} - \frac{ax}{x} = \frac{a - ax}{x} = \frac{a(1 - x)}{x}$$ Rewrite denominator: $$x + \frac{y}{x} = \frac{x^2}{x} + \frac{y}{x} = \frac{x^2 + y}{x}$$ Expression becomes: $$\frac{\frac{a(1 - x)}{x}}{\frac{x^2 + y}{x}} = \frac{a(1 - x)}{x} \times \frac{x}{x^2 + y} = \frac{a(1 - x)}{x^2 + y}$$ --- **Final answers:** - $x = -3$ and $(x^2 - 5)(x^3 - x) = -96$ - Factored quadratics as shown in steps 6-14 and 16-24 - Solutions to systems: (2,3), (1,1), (3,-4), (4,5) - Simplified radicals: $14\sqrt{5} - 20\sqrt{2}$ and $16\sqrt{2}$ - Simplified algebraic fractions: $y + 1$, $\frac{a - 4b}{x - b^2}$, $\frac{a(1 - x)}{x^2 + y}$