Solutions to Try Its

1. [latex]\left(x - 6\right)\left(x+1\right)=0;x=6,x=-1[/latex]

2. [latex]\left(x - 7\right)\left(x+3\right)=0[/latex], [latex]x=7[/latex], [latex]x=-3[/latex].

3. [latex]\left(x+5\right)\left(x - 5\right)=0[/latex], [latex]x=-5[/latex], [latex]x=5[/latex].

4. [latex]\left(3x+2\right)\left(4x+1\right)=0[/latex], [latex]x=-\frac{2}{3}[/latex], [latex]x=-\frac{1}{4}[/latex]

5. [latex]x=0,x=-10,x=-1[/latex]

6. [latex]x=4\pm \sqrt{5}[/latex]

7. [latex]x=3\pm \sqrt{22}[/latex]

8. [latex]x=-\frac{2}{3}[/latex], [latex]x=\frac{1}{3}[/latex]

9. [latex]5[/latex] 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 [latex]a\cdot b=0[/latex] then it must follow that each factor separately offers a solution to the product being zero: [latex]a=0\text{ }or\text{ b}=0[/latex].

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

7. [latex]x=6[/latex], [latex]x=3[/latex]

9. [latex]x=\frac{-5}{2}[/latex], [latex]x=\frac{-1}{3}[/latex]

11. [latex]x=5[/latex], [latex]x=-5[/latex]

13. [latex]x=\frac{-3}{2}[/latex], [latex]x=\frac{3}{2}[/latex]

15. [latex]x=-2[/latex]

17. [latex]x=0[/latex], [latex]x=\frac{-3}{7}[/latex]

19. [latex]x=-6[/latex], [latex]x=6[/latex]

21. [latex]x=6[/latex], [latex]x=-4[/latex]

23. [latex]x=1[/latex], [latex]x=-2[/latex]

25. [latex]x=-2[/latex], [latex]x=11[/latex]

27. [latex]x=3\pm \sqrt{22}[/latex]

29. [latex]z=\frac{2}{3}\\[/latex], [latex]z=-\frac{1}{2}[/latex]

31. [latex]x=\frac{3\pm \sqrt{17}}{4}[/latex]

33. Not real

35. One rational

37. Two real; rational

39. [latex]x=\frac{-1\pm \sqrt{17}}{2}[/latex]

41. [latex]x=\frac{5\pm \sqrt{13}}{6}[/latex]

43. [latex]x=\frac{-1\pm \sqrt{17}}{8}[/latex]

45. [latex]x\approx 0.131[/latex] and [latex]x\approx 2.535[/latex]

47. [latex]x\approx -6.7[/latex] and [latex]x\approx 1.7[/latex]

49. [latex]\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}[/latex]

51. [latex]x\left(x+10\right)=119[/latex]; 7 ft. and 17 ft.

53. maximum at [latex]x=70[/latex]

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

57. 3 feet