2.)
[tex]3s(s - 2) = 12s \\ 3 {s}^{2} - 6s = 12s \\ 3 {s}^{2} - 6s - 12s = 0 \\ 3 {s}^{2} - 18s = 0 \\ {s}^{2} - 6s = 0 \\ s(s - 6) = 0 \\ s = \frac{0}{s - 6} \: \: \: \: \: s = 0 \\ or \\ s - 6 = \frac{0}{s} \\ s - 6 = 0 \\ s = 6[/tex]
3.)
[tex] {(t + 1)}^{2} + {(t - 8)}^{2} = 45 \\ ( {t}^{2} + t + t + 1) + ( {t}^{2} - 8t - 8t + 64) = 45 \\ {t}^{2} + 2t + 1 + {t}^{2} - 16t + 64 = 45 \\ 2 {t}^{2} - 14t + 65 = 45 \\ 2 {t}^{2} - 14t = 45 - 65 \\ 2 {t}^{2} - 14t = - 20 \\ {t}^{2} - 7t + 10 = 0 \\ (t - 2)(t - 5) = 0 \\ t - 2 = 0 \: \: \: t - 5 = 0 \\ t = 2 \\ t = 5[/tex]
7.)
[tex] \frac{4}{t - 3} + \frac{t}{2} = - 2 \\ \frac{2(4) + t(t - 3)}{(t - 3)(2)} = - 2 \\ \frac{8 + {t}^{2} - 3t }{2t - 6} = - 2 \\ 8 + {t}^{2} - 3t = ( - 2)(2t - 6) \\ 8 + {t}^{2} - 3t = - 4t + 12 \\ {t}^{2} - 3t + 4t + 8 - 12 = 0 \\ {t}^{2} + t - 4 = 0 \\ \\ use \: quadratic \: formula \: to \: find \: the \: value \: of \: t[/tex]
10.)
[tex] \frac{2x}{x - 5} + \frac{1}{x - 3} = 3 \\ \frac{2x(x - 3) + 1(x - 5)}{(x - 5)(x - 3)} = 3 \\ \frac{2 {x}^{2} - 6x + x - 5 }{ {x}^{2} - 3x - 5x + 15} = 3 \\ \frac{2 {x}^{2} - 5x - 5 }{ {x}^{2} - 8x + 15} = 3 \\ 2 {x}^{2} - 5x - 5 = 3( {x}^{2} - 8x + 15) \\ 2 {x}^{2} - 5x - 5 = 3 {x}^{2} - 24x + 45 \\ 2 {x}^{2} - 3 {x }^{2} - 5x + 24x - 5 - 45 = 0 \\ - {x}^{2} + 19x - 50 = 0 \\ use \: quadratic \: formula \: to \: find \: the \: value \: of \: x[/tex]
