Solutions to Try Its

1. $\left(x - 6\right)\left(x+1\right)=0;x=6,x=-1$

2. $\left(x - 7\right)\left(x+3\right)=0$, $x=7$, $x=-3$.

3. $\left(x+5\right)\left(x - 5\right)=0$, $x=-5$, $x=5$.

4. $\left(3x+2\right)\left(4x+1\right)=0$, $x=-\frac{2}{3}$, $x=-\frac{1}{4}$

5. $x=0,x=-10,x=-1$

6. $x=4\pm \sqrt{5}$

7. $x=3\pm \sqrt{22}$

8. $x=-\frac{2}{3}$, $x=\frac{1}{3}$

9. $5$ units

Solutions to Odd-Numbered Exercises

1. It is a second-degree equation (the highest variable exponent is 2).

3. We want to take advantage of the zero property of multiplication in the fact that if $a\cdot b=0$ then it must follow that each factor separately offers a solution to the product being zero: $a=0\text{ }or\text{ b}=0$.

5. One, when no linear term is present (no x term), such as ${x}^{2}=16$. Two, when the equation is already in the form ${\left(ax+b\right)}^{2}=d$.

7. $x=6$, $x=3$

9. $x=\frac{-5}{2}$, $x=\frac{-1}{3}$

11. $x=5$, $x=-5$

13. $x=\frac{-3}{2}$, $x=\frac{3}{2}$

15. $x=-2$

17. $x=0$, $x=\frac{-3}{7}$

19. $x=-6$, $x=6$

21. $x=6$, $x=-4$

23. $x=1$, $x=-2$

25. $x=-2$, $x=11$

27. $x=3\pm \sqrt{22}$

29. $z=\frac{2}{3}\\$, $z=-\frac{1}{2}$

31. $x=\frac{3\pm \sqrt{17}}{4}$

33. Not real

35. One rational

37. Two real; rational

39. $x=\frac{-1\pm \sqrt{17}}{2}$

41. $x=\frac{5\pm \sqrt{13}}{6}$

43. $x=\frac{-1\pm \sqrt{17}}{8}$

45. $x\approx 0.131$ and $x\approx 2.535$

47. $x\approx -6.7$ and $x\approx 1.7$

49. $\begin{array}{l}a{x}^{2}+bx+c \hfill& =0\hfill \\ {x}^{2}+\frac{b}{a}x \hfill& =\frac{-c}{a}\hfill \\ {x}^{2}+\frac{b}{a}x+\frac{{b}^{2}}{4{a}^{2}} \hfill& =\frac{-c}{a}+\frac{b}{4{a}^{2}}\hfill \\ {\left(x+\frac{b}{2a}\right)}^{2}\hfill& =\frac{{b}^{2}-4ac}{4{a}^{2}}\hfill \\ x+\frac{b}{2a}\hfill& =\pm \sqrt{\frac{{b}^{2}-4ac}{4{a}^{2}}}\hfill \\ x\hfill& =\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}\hfill \end{array}$

51. $x\left(x+10\right)=119$; 7 ft. and 17 ft.

53. maximum at $x=70$

55. The quadratic equation would be $\left(100x - 0.5{x}^{2}\right)-\left(60x+300\right)=300$. The two values of $x$ are 20 and 60.

57. 3 feet