(a)

\begin{align} x^2 & + 4x + 1 = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 4 \pm \sqrt{(4)^2 - 4(1)(1)} \over 2(1) } \\ & = { - 4 \pm \sqrt{12} \over 2} \\ & = -0.26794 \text{ or } -3.7320 \\ & \approx -0.268 \text{ or } -3.73 \end{align}

(b)

\begin{align} 3x^2 & + 6x - 1 = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 6 \pm \sqrt{(6)^2 - 4(3)(-1)} \over 2(3) } \\ & = { - 6 \pm \sqrt{48} \over 6} \\ & = 0.15470 \text{ or } -2.1547 \\ & \approx 0.155 \text{ or } -2.15 \end{align}

(c)

\begin{align} 3x^2 & - 5x - 17 = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-5) \pm \sqrt{(-5)^2 - 4(3)(-17)} \over 2(3) } \\ & = { 5 \pm \sqrt{229} \over 6} \\ & = 3.3554 \text{ or } -1.6887 \\ & \approx 3.36 \text{ or } -1.69 \end{align}

(d)

\begin{align} -3x^2 & - 7x + 9 = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-7) \pm \sqrt{(-7)^2 - 4(-3)(9)} \over 2(-3) } \\ & = { 7 \pm \sqrt{157} \over -6} \\ & = -3.2549 \text{ or } 0.92166 \\ & \approx -3.25 \text{ or } 0.922 \end{align}

(e)

\begin{align} 2 + 2x^2 - 7x & = 0 \\ 2x^2 - 7x + 2 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-7) \pm \sqrt{(-7)^2 - 4(2)(2)} \over 2(2) } \\ & = { 7 \pm \sqrt{33} \over 4} \\ & = 3.1861 \text{ or } 0.31385 \\ & \approx 3.19 \text{ or } 0.314 \end{align}

(f)

\begin{align} 10x - 5x^2 - 2 & = 0 \\ - 5x^2 + 10x - 2 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 10 \pm \sqrt{(10)^2 - 4(-5)(-2)} \over 2(-5) } \\ & = { -10 \pm \sqrt{60} \over -10} \\ & = 0.22540 \text{ or } 1.7745 \\ & \approx 0.225 \text{ or } 1.77 \end{align}

(a)

\begin{align} x^2 + 5x & = 21 \\ x^2 + 5x - 21 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 5 \pm \sqrt{(5)^2 - 4(1)(-21)} \over 2(1) } \\ & = { -5 \pm \sqrt{109} \over 2} \\ & = 2.7201 \text{ or } -7.7201 \\ & \approx 2.72 \text{ or } -7.72 \end{align}

(b)

\begin{align} 10x^2 - 12x & = 15 \\ 10x^2 - 12x - 15 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-12) \pm \sqrt{(-12)^2 - 4(10)(-15)} \over 2(10) } \\ & = { 12 \pm \sqrt{744} \over 20} \\ & = 1.9638 \text{ or } -0.76381 \\ & \approx 1.96 \text{ or } -0.764 \end{align}

(c)

\begin{align} 8x^2 & = 3x + 6 \\ 8x^2 - 3x - 6 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-3) \pm \sqrt{(-3)^2 - 4(8)(-6)} \over 2(8) } \\ & = { 3 \pm \sqrt{201} \over 16} \\ & = 1.0735 \text{ or } -0.69859 \\ & \approx 1.07 \text{ or } -0.699 \end{align}

(d)

\begin{align} 4x^2 + 1 & = - 4x \\ 4x^2 + 4x + 1 & = 0 \\ (2x + 1)(2x + 1) & = 0 \\ (2x + 1)^2 & = 0 \\ 2x + 1 & = \sqrt{0} \\ 2x + 1 & = 0 \\ 2x & = -1 \\ x & = {-1 \over 2} \\ x & = -0.5 \end{align}

(e)

\begin{align} 9 - 5x^2 & = -3x \\ 0 & = 5x^2 - 3x - 9 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-3) \pm \sqrt{(-3)^2 - 4(5)(-9)} \over 2(5) } \\ & = { 3 \pm \sqrt{189} \over 10} \\ & = 1.6747 \text{ or } -1.0747 \\ & \approx 1.67 \text{ or } -1.07 \end{align}

(f)

\begin{align} 16x - 61 & = x^2 \\ 0 & = x^2 - 16x + 61 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-16) \pm \sqrt{(-16)^2 - 4(1)(61)} \over 2(1) } \\ & = { 16 \pm \sqrt{12} \over 2} \\ & = 9.7320 \text{ or } 6.2679 \\ & \approx 9.73 \text{ or } 6.27 \end{align}

(a)

\begin{align} {8 \over x} & = 2x + 1 \\ {8 \over x} & = {2x + 1 \over 1} \\ 8 & = x(2x + 1) \\ 8 & = 2x^2 + x \\ 0 & = 2x^2 + x - 8 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 1 \pm \sqrt{(1)^2 - 4(2)(-8)} \over 2(2) } \\ & = { -1 \pm \sqrt{65} \over 4} \\ & = 1.7655 \text{ or } -2.2655 \\ & \approx 1.77 \text{ or } -2.27 \end{align}

(b)

\begin{align} x + {7 \over x} & = 9 \\ {7 \over x} & = 9 - x \\ {7 \over x} & = {9 - x \over 1} \\ 7 & = x(9 - x) \\ 7 & = 9x - x^2 \\ 0 & = -x^2 + 9x - 7 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 9 \pm \sqrt{(9)^2 - 4(-1)(-7)} \over 2(-1) } \\ & = { -9 \pm \sqrt{53} \over -2} \\ & = 0.85994 \text{ or } 8.1400 \\ & \approx 0.860 \text{ or } 8.14 \end{align}

(c)

\begin{align} {x + 1 \over 5 - x} & = x \\ {x + 1 \over 5 - x} & = {x \over 1} \\ x + 1 & = x(5 - x) \\ x + 1 & = 5x - x^2 \\ 0 & = - x^2 + 5x - x - 1 \\ 0 & = - x^2 + 4x - 1 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 4 \pm \sqrt{(4)^2 - 4(-1)(-1)} \over 2(-1) } \\ & = { -4 \pm \sqrt{12} \over -2} \\ & = 0.26794 \text{ or } 3.7320 \\ & \approx 0.268 \text{ or } 3.73 \end{align}

(d)

\begin{align} {3x - 1} & = -{9 \over 3x + 5} \\ {3x - 1 \over 1} & = {-9 \over 3x + 5} \\ (3x - 1)(3x + 5) & = - 9 \\ 9x^2 + 15x - 3x - 5 & = - 9 \\ 9x^2 + 12x - 5 + 9 & = 0 \\ 9x^2 + 12x + 4 & = 0 \\ (3x + 2)(3x + 2) & = 0 \\ (3x + 2)^2 & = 0 \\ 3x + 2 & = \sqrt{0} \\ 3x + 2 & = 0 \\ 3x & = -2 \\ x & = -{2 \over 3} \end{align}

(e)

\begin{align} 2x + 1 & = {x + 1 \over x - 5} \\ {2x + 1 \over 1} & = {x + 1 \over x - 5} \\ (2x + 1)(x - 5) & = x + 1 \\ 2x^2 - 10x + x - 5 & = x + 1 \\ 2x^2 - 9x - 5 & = x + 1 \\ 2x^2 - 9x - x - 5 - 1 & = 0 \\ 2x^2 - 10x - 6 & = 0 \\ x^2 - 5x - 3 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-5) \pm \sqrt{(-5)^2 - 4(1)(-3)} \over 2(1) } \\ & = { 5 \pm \sqrt{37} \over 2} \\ & = 5.5413 \text{ or } -0.54138 \\ & \approx 5.54 \text{ or } -0.541 \end{align}

(f)

\begin{align} {5x \over x + 4} & = 4x + 1 \\ {5x \over x + 4} & = {4x + 1 \over 1} \\ 5x & = (4x + 1)(x + 4) \\ 5x & = 4x^2 + 16x + x + 4 \\ 5x & = 4x^2 + 17x + 4 \\ 0 & = 4x^2 + 17x - 5x + 4 \\ 0 & = 4x^2 + 12x + 4 \\ 0 & = x^2 + 3x + 1 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 3 \pm \sqrt{(3)^2 - 4(1)(1)} \over 2(1) } \\ & = { -3 \pm \sqrt{5} \over 2} \\ & = -0.38196 \text{ or } -2.6180 \\ & \approx -0.382 \text{ or } -2.62 \end{align}

(a)

\begin{align} x(x + 1) & = 1 \\ x^2 + x & = 1 \\ x^2 + x - 1 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 1 \pm \sqrt{(1)^2 - 4(1)(-1)} \over 2(1) } \\ & = { -1 \pm \sqrt{5} \over 2} \\ & = 0.61803 \text{ or } -1.6180 \\ & \approx 0.618 \text{ or } -1.62 \end{align}

(b)

\begin{align} 3(x + 1)(x - 1) & = 7x \\ 3(x^2 - 1^2) & = 7x \phantom{000000} [(a + b)(a - b) = a^2 - b^2] \\ 3(x^2 - 1) & = 7x \\ 3x^2 - 3 & = 7x \\ 3x^2 - 7x - 3 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-7) \pm \sqrt{(-7)^2 - 4(3)(-3)} \over 2(3) } \\ & = { 7 \pm \sqrt{85} \over 6} \\ & = 2.7032 \text{ or } -0.36992 \\ & \approx 2.70 \text{ or } -0.370 \end{align}

(c)

\begin{align} (x - 1)^2 - 2x & = 0 \\ (x)^2 - 2(x)(1) + (1)^2 - 2x & = 0 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ x^2 - 2x + 1 - 2x & = 0 \\ x^2 - 4x + 1 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-4) \pm \sqrt{(-4)^2 - 4(1)(1)} \over 2(1) } \\ & = { 4 \pm \sqrt{12} \over 2} \\ & = 3.7320 \text{ or } 0.26794 \\ & \approx 3.73 \text{ or } 0.268 \end{align}

(d)

\begin{align} x(x - 5) & = 7 - 2x \\ x^2 - 5x & = 7 - 2x \\ x^2 - 5x + 2x - 7 & = 0 \\ x^2 - 3x - 7 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-3) \pm \sqrt{(-3)^2 - 4(1)(-7)} \over 2(1) } \\ & = { 3 \pm \sqrt{37} \over 2} \\ & = 4.5413 \text{ or } -1.5413 \\ & \approx 4.54 \text{ or } -1.54 \end{align}

(e)

\begin{align} (5x - 9)(x - 1) - x(x - 2) & = 0 \\ 5x^2 - 5x - 9x + 9 - x^2 + 2x & = 0 \\ 5x^2 - x^2 - 5x - 9x + 2x + 9 & = 0 \\ 4x^2 - 12x + 9 & = 0 \\ (2x - 3)(2x - 3) & = 0 \\ (2x - 3)^2 & = 0 \\ 2x - 3 & = \sqrt{0} \\ 2x - 3 & = 0 \\ 2x & = 3 \\ x & = {3 \over 2} \\ x & = 1.5 \end{align}

(f)

\begin{align} (4x - 3)^2 + (4x + 3)^2 & = 25 \\ \underbrace{(4x)^2 - 2(4x)(3) + 3^2}_{(a - b)^2 = a^2 - 2ab + b^2} + \underbrace{(4x)^2 + 2(4x)(3) + (3)^2}_{(a + b)^2 = a^2 + 2ab + b^2} & = 25 \\ \\ 16x^2 - 24x + 9 + 16x^2 + 24x + 9 & = 25 \\ 32x^2 + 18 & = 25 \\ 32x^2 & = 25 - 18 \\ 32x^2 & = 7 \\ x^2 & = {7 \over 32} \\ x & = \pm \sqrt{7 \over 32} \\ x & = \pm 0.46770 \\ x & \approx \pm 0.468 \end{align}

(a)

\begin{align} {x - 1 \over x + 1} & = {8x \over 1 - x} \\ (x - 1)(1 - x) & = 8x(x + 1) \\ x - x^2 - 1 + x & = 8x^2 + 8x \\ 2x - x^2 - 1 & = 8x^2 + 8x \\ 0 & = 8x^2 + x^2 + 8x - 2x + 1 \\ 0 & = 9x^2 + 6x + 1 \\ 0 & = (3x + 1)(3x + 1) \\ 0 & = (3x + 1)^2 \\ \sqrt{0} & = 3x + 1 \\ 0 & = 3x + 1 \\ -1 & = 3x \\ -{1 \over 3} & = x \end{align}

(b)

\begin{align} { (x - 2)(x - 3) \over (x - 1)(x + 2)} & = {2 \over 3} \\ { x^2 - 3x - 2x + 6 \over x^2 + 2x - x - 2 } & = {2 \over 3} \\ { x^2 - 5x + 6 \over x^2 + x - 2} & = {2 \over 3} \\ 3(x^2 - 5x + 6) & = 2(x^2 + x - 2) \\ 3x^2 - 15x + 18 & = 2x^2 + 2x - 4 \\ 3x^2 - 2x^2 - 15x - 2x + 18 + 4 & = 0 \\ x^2 - 17x + 22 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-17) \pm \sqrt{(-17)^2 - 4(1)(22)} \over 2(1) } \\ & = { 17 \pm \sqrt{201} \over 2} \\ & = 15.588 \text{ or } 1.4112 \\ & \approx 15.6 \text{ or } 1.41 \end{align}

\begin{align} {x(x - 3) \over (x + 1)^2} & = {3 \over 5} \\ {x^2 - 3x \over x^2 + 2(x)(1) + (1)^2} & = {3 \over 5} \phantom{000000} [(a + b)^2 = a^2 + 2ab + b^2] \\ {x^2 - 3x \over x^2 + 2x + 1} & = {3 \over 5} \\ 5(x^2 - 3x) & = 3(x^2 + 2x + 1) \\ 5x^2 - 15x & = 3x^2 + 6x + 3 \\ 5x^2 - 3x^2 - 15x - 6x - 3 & = 0 \\ 2x^2 - 21x - 3 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-21) \pm \sqrt{(-21)^2 - 4(2)(-3)} \over 2(2) } \\ & = { 21 \pm \sqrt{465} \over 4} \\ & = 10.640 \text{ or } -0.14096 \\ & \approx 10.6 \text{ or } -0.141 \end{align}

(a)

\begin{align} {x \over 2} & = {4 \over x} - 1 \\ {x \over 2} & = {4 \over x} - {x \over x} \\ {x \over 2} & = {4 - x \over x} \\ x(x) & = 2(4 - x) \\ x^2 & = 8 - 2x \\ x^2 + 2x - 8 & = 0 \\ (x + 4)(x - 2) & = 0 \end{align} \begin{align} x + 4 & = 0 && \text{ or } & x - 2 & = 0 \\ x & = -4 &&& x & = 2 \end{align}

(b)

\begin{align} {2 \over x + 5} & = 1 - {x + 1 \over 5} \\ {2 \over x + 5} & = {5 \over 5} - {x + 1 \over 5} \\ {2 \over x + 5} & = {5 - (x + 1) \over 5} \\ {2 \over x + 5} & = {5 - x - 1 \over 5} \\ {2 \over x + 5} & = {4 - x \over 5} \\ 5(2) & = (x + 5)(4 - x) \\ 10 & = 4x - x^2 + 20 - 5x \\ 10 & = - x^2 - x + 20 \\ 0 & = - x^2 - x + 20 - 10 \\ 0 & = - x^2 - x + 10 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-1) \pm \sqrt{(-1)^2 - 4(-1)(10)} \over 2(-1) } \\ & = { 1 \pm \sqrt{41} \over -2} \\ & = -3.7015 \text{ or } 2.7015 \\ & \approx -3.70 \text{ or } 2.70 \end{align}

(c)

\begin{align} {x - 2 \over 5} + {1 \over 2x - 3} & = 2 \\ {(x - 2)(2x - 3) \over 5(2x - 3)} + {5 \over 5(2x - 3)} & = 2 \\ {(x - 2)(2x - 3) + 5 \over 5(2x - 3)} & = 2 \\ {2x^2 - 3x - 4x + 6 + 5 \over 10x - 15} & = 2 \\ {2x^2 - 7x + 11 \over 10x - 15} & = {2 \over 1} \\ 2x^2 - 7x + 11 & = 2(10x - 15) \\ 2x^2 - 7x + 11 & = 20x - 30 \\ 2x^2 - 7x - 20x + 11 + 30 & = 0 \\ 2x^2 - 27x + 41 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-27) \pm \sqrt{(-27)^2 - 4(2)(41)} \over 2(2) } \\ & = { 27 \pm \sqrt{401} \over 4} \\ & = 11.756 \text{ or } 1.7437 \\ & \approx 11.8 \text{ or } 1.74 \end{align}

(d)

\begin{align} {4 \over x} - {1 \over x - 1} & = 9 \\ {4(x - 1) \over x(x - 1)} - {x \over x(x - 1)} & = 9 \\ {4(x - 1) - x \over x(x - 1)} & = 9 \\ {4x - 4 - x \over x^2 - x} & = 9 \\ {3x - 4 \over x^2 - x} & = {9 \over 1} \\ 3x - 4 & = 9(x^2 - x) \\ 3x - 4 & = 9x^2 - 9x \\ 0 & = 9x^2 - 9x - 3x + 4 \\ 0 & = 9x^2 - 12x + 4 \\ 0 & = (3x - 2)(3x - 2) \\ 0 & = (3x - 2)^2 \\ \sqrt{0} & = 3x - 2 \\ 0 & = 3x - 2 \\ 2 & = 3x \\ {2 \over 3} & = x \end{align}

(e)

\begin{align} {1 \over x + 2} + {1 \over x - 2} & = {3 \over 11} \\ {x - 2 \over (x + 2)(x - 2)} + {x + 2 \over (x + 2)(x - 2)} & = {3 \over 11} \\ {x - 2 + x + 2 \over (x + 2)(x - 2)} & = {3 \over 11} \\ {2x \over x^2 - 2^2} & = {3 \over 11} \phantom{000000} [(a + b)(a - b) = a^2 - b^2] \\ {2x \over x^2 - 4} & = {3 \over 11} \\ 11(2x) & = 3(x^2 - 4) \\ 22x & = 3x^2 - 12 \\ 0 & = 3x^2 - 22x - 12 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-22) \pm \sqrt{(-22)^2 - 4(3)(-12)} \over 2(3) } \\ & = { 22 \pm \sqrt{628} \over 6} \\ & = 7.8433 \text{ or } -0.50998 \\ & \approx 7.84 \text{ or } -0.510 \end{align}

(f)

\begin{align} {7 \over x - 1} - {x + 1 \over x + 3} & = {1 \over 2} \\ {7(x + 3) \over (x - 1)(x + 3)} - {(x + 1)(x - 1) \over (x - 1)(x + 3)} & = {1 \over 2} \\ {7(x + 3) - (x + 1)(x - 1) \over (x - 1)(x + 3)} & = {1 \over 2} \\ {7x + 21 - (x^2 - 1^2) \over x^2 + 3x - x - 3 } & = {1 \over 2} \phantom{000000} [(a + b)(a - b) = a^2 - b^2] \\ {7x + 21 - x^2 + 1 \over x^2 + 2x - 3} & = {1 \over 2} \\ {-x^2 + 7x + 22 \over x^2 + 2x - 3} & = {1 \over 2} \\ 2(-x^2 + 7x + 22) & = x^2 + 2x - 3 \\ -2x^2 + 14x + 44 & = x^2 + 2x - 3 \\ 0 & = x^2 + 2x^2 + 2x - 14x - 3 - 44 \\ 0 & = 3x^2 - 12x - 47 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-12) \pm \sqrt{(-12)^2 - 4(3)(-47)} \over 2(3) } \\ & = { 12 \pm \sqrt{708} \over 6} \\ & = 6.4347 \text{ or } -2.4347 \\ & \approx 6.43 \text{ or } -2.43 \end{align}

(g)

\begin{align} {5 \over x - 2} & = 2 - {4 \over (x - 2)^2} \\ {5 \over x - 2} + {4 \over (x - 2)^2} & = 2 \\ {5(x - 2) \over (x - 2)^2} + {4 \over (x - 2)^2} & = 2 \\ {5(x - 2) + 4 \over (x - 2)^2} & = 2 \\ {5x - 10 + 4 \over (x)^2 - 2(x)(2) + (2)^2} & = 2 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ {5x - 6 \over x^2 - 4x + 4} & = {2 \over 1} \\ 5x - 6 & = 2(x^2 - 4x + 4) \\ 5x - 6 & = 2x^2 - 8x + 8 \\ 0 & = 2x^2 - 8x - 5x + 8 + 6 \\ 0 & = 2x^2 - 13x + 14 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-13) \pm \sqrt{(-13)^2 - 4(2)(14)} \over 2(2) } \\ & = { 13 \pm \sqrt{57} \over 4} \\ & = 5.1374 \text{ or } 1.3625 \\ & \approx 5.14 \text{ or } 1.36 \end{align}

(h)

\begin{align} {5 \over x - 1} + {x \over (x - 1)^2} & = 1 \\ {5(x - 1) \over (x - 1)^2} + {x \over (x - 1)^2} & = 1 \\ {5(x - 1) + x \over (x - 1)^2} & = 1 \\ {5x - 5 + x \over (x)^2 - 2(x)(1) + (1)^2} & = 1 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ {6x -5 \over x^2 - 2x + 1} & = {1 \over 1} \\ 6x - 5 & = x^2 - 2x + 1 \\ 0 & = x^2 - 2x - 6x + 1 + 5 \\ 0 & = x^2 - 8x + 6 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-8) \pm \sqrt{(-8)^2 - 4(1)(6)} \over 2(1) } \\ & = { 8 \pm \sqrt{40} \over 2} \\ & = 7.1622 \text{ or } 0.83772 \\ & \approx 7.16 \text{ or } 0.838 \end{align}

(a)

\begin{align} {1 \over x} + {2 \over x - 1} + {3 \over x + 1} & = 0 \\ {2 \over x - 1} + {3 \over x + 1} & = - {1 \over x} \\ {2(x + 1) \over (x + 1)(x - 1)} + {3(x - 1) \over (x + 1)(x -1 )} & = -{1 \over x} \\ {2(x + 1) + 3(x - 1) \over (x + 1)(x - 1)} & = -{1 \over x} \\ {2x + 2 + 3x - 3 \over x^2 - 1^2} & = -{1 \over x} \phantom{000000} [(a + b)(a - b) = a^2 - b^2] \\ {5x - 1 \over x^2 - 1} & = {-1 \over x} \\ x(5x - 1) & = (-1)(x^2 - 1) \\ 5x^2 - x & = - x^2 + 1 \\ 5x^2 + x^2 - x - 1 & = 0 \\ 6x^2 - x - 1 & = 0 \\ (2x - 1)(3x + 1) & = 0 \end{align} \begin{align} 2x - 1 & = 0 && \text{ or } & 3x + 1 & =0 \\ 2x & = 1 &&& 3x & = -1 \\ x & = {1 \over 2} &&& x & = -{1 \over 3} \end{align}

(b)

\begin{align} {1 \over x^2 - 9} - {2 \over 3 - x} & = 1 \\ {1 \over (x + 3)(x - 3)} - {2 \over 3 - x} & = 1 \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \\ {1 \over (x + 3)(x - 3)} + {2 \over x - 3} & = 1 \\ {1 \over (x + 3)(x - 3)} + {2(x + 3) \over (x + 3)(x - 3)} & = 1 \\ {1 + 2(x + 3) \over (x + 3)(x - 3)} & = 1 \\ {1 + 2x + 6 \over x^2 - 9} & = 1 \\ {2x + 7 \over x^2 - 9} & = {1 \over 1} \\ 2x + 7 & = x^2 - 9 \\ 0 & = x^2 - 2x - 9 - 7 \\ 0 & = x^2 - 2x - 16 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-2) \pm \sqrt{(-2)^2 - 4(1)(-16)} \over 2(1) } \\ & = { 2 \pm \sqrt{68} \over 2} \\ & = 5.1231 \text{ or } -3.1231 \\ & \approx 5.12 \text{ or } -3.12 \end{align}

(c)

\begin{align} {3 \over x - 3} + {x + 1 \over x^2 - 5x + 6} & = 1 \\ {3 \over x - 3} + {x + 1 \over (x - 3)(x - 2)} & = 1 \\ {3(x - 2) \over (x - 3)(x - 2)} + {x + 1 \over (x - 3)(x - 2)} & = 1 \\ {3(x - 2) + x + 1 \over (x - 3)(x - 2)} & = 1 \\ {3x - 6 + x + 1 \over x^2 - 5x + 6} & = 1 \\ {4x - 5 \over x^2 - 5x + 6} & = {1 \over 1} \\ 4x - 5 & = x^2 - 5x + 6 \\ 0 & = x^2 - 5x - 4x + 6 + 5 \\ 0 & = x^2 - 9x + 11 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-9) \pm \sqrt{(-9)^2 - 4(1)(11)} \over 2(1) } \\ & = { 9 \pm \sqrt{37} \over 2} \\ & = 7.5413 \text{ or } 1.4586 \\ & \approx 7.54 \text{ or } 1.46 \end{align}

(d)

\begin{align} {4 \over x - 1} + 2 & = {x + 2 \over 2x^2 + 3x - 5} \\ {4 \over x - 1} + 2 & = {x + 2 \over (x - 1)(2x + 5)} \\ 2 & = {x + 2 \over (x - 1)(2x + 5)} - {4 \over x - 1} \\ 2 & = {x + 2 \over (x - 1)(2x + 5)} - {4(2x + 5) \over (x - 1)(2x + 5)} \\ 2 & = {x + 2 - 4(2x + 5) \over (x - 1)(2x + 5)} \\ 2 & = {x + 2 - 8x - 20 \over (x - 1)(2x + 5)} \\ {2 \over 1} & = {- 7x - 18 \over 2x^2 + 3x - 5} \\ 2(2x^2 + 3x - 5) & = - 7x - 18 \\ 4x^2 + 6x - 10 & = - 7x - 18 \\ 4x^2 + 6x + 7x - 10 + 18 & = 0 \\ 4x^2 + 13x + 8 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 13 \pm \sqrt{(13)^2 - 4(4)(8)} \over 2(4) } \\ & = { -13 \pm \sqrt{41} \over 8} \\ & = -0.82460 \text{ or } -2.4253 \\ & \approx -0.825 \text{ or } -2.43 \end{align}