Question

1/x-1 - 3/x+2 = 1/4

solve to find x

Answer 1

`(1)/(x-1) - (3)/(x+2) = (1)/(4) \ /\cdot 4(x-1)(x+2)\\\\(4(x-1)(x+2))/(x-1) - (3\cdot 4(x-1)(x+2))/(x+2) = (4(x-1)(x+2))/(4)\\\\4(x+2)-12(x-1)=(x-1)(x+2)\\\\4x+8-12x+12=x^2+2x-x-2\\\\-8x+20=x^2+x-2\\\\-x^2-9x+22=0\ /\cdot(-1)\\\\x^2+9x-22=0\ \ \ \Rightarrow\ \ \ x^2+11x-2x-22=0\\\\x(x+11)-2(x+11)=0\\\\(x+11)(x-2)=0\ \ \ \ \ \Leftrightarrow\ \ \ (x+11=0\ \ \ or\ \ \ x-2=0)\\\\x=-11\ \ \ or\ \ \ x=2`

Answer 2

`(1)/(x-1)-(3)/(x+2)=(1)/(4)\\ \\ (x+2-3(x-1))/((x-1)(x+2))=(1)/(4)\\ \\ (x+2-3x+3)/((x-1)(x+2))=(1)/(4)\\ \\ (5-2x)/(x^2+x-2)=(1)/(4)\\ \\ 4(5-2x)=x^2+x-2\\ \\ 20-8x=x^2+x-2\\ \\ \boxed{x^2+9x-22=0}`

Solvin this quadratic equation with Bhaskara formula we find:

x = -11 or x = 2