[tex]a.\: Area \: of \: carpet: 24 \times 30 = 720 \: {ft}^{2} \\ Area \: of \:border:720 - 432 = 288 \: {ft}^{2} \\ Equation:60x + 2x(24 - 2x) = 288 \: (0 < x < 24) \\ \\ b. \: 60x + 2x(24 - 2x) = 288 \\ \Leftrightarrow - 4 {x}^{2} + 48x + 60x = 288 \\ \Leftrightarrow - 4 {x}^{2} + 108x - 288 = 0 \\ \Leftrightarrow {x}^{2} - 27x + 72 = 0 \\ \Leftrightarrow {x}^{2} - 3x - 24x + 72 = 0 \\ \Leftrightarrow x(x - 3) - 24(x - 3) = 0 \\ \Leftrightarrow (x - 24)(x - 3) = 0 \\ \Leftrightarrow x - 24 = 0 \: \vee \: x - 3 = 0 \\ \Leftrightarrow x = 24 \: \vee \:x = 3 \\ \Rightarrow x = 3 \: (0 < x < 24) \\ \Rightarrow The \: width \: of \: the \: border \: is \: 3 \: ft[/tex]
