Question

Find the roots of the equation 1 upon x - 2+1 upon x =8 upon 2x+5,x is not equal to the 0,2,-5upon2

Answer 1

$\frac{1}{x - 2} + \frac{1}{x} = \frac{8}{2x + 5} \\ \frac{x + (x - 2)}{x(x - 2)} = \frac{8}{2x + 5} \\ (2x - 2)(2x + 5) = 8x(x - 2) \\ 4 {x}^{2} + 6x - 10 = 8 {x}^{2} - 16 x\\ 4 {x}^{2} - 22x + 10 = 0 \\ 2{x}^{2} - 11x + 5 = 0 \\ 2 {x}^{2} - 10x - x + 5 = 0 \\ 2x(x - 5) - (x - 5) = 0 \\ (2x - 1)(x - 5) = 0 \\ x = \frac{1}{2} \: \: or \: \: 5$