Question

The reciprocal of an integer plus the reciprocal of two times the integer plus two equals 2/3   Find the integer.

Answer 1

`(1)/(x) + (1)/(2x+2)= (2)/(3)\ /\cdot 3x(2x+2) \ \ \ \wedge\ \ \ x \\eq 0\ \ \wedge\ \ 2x+2 \\eq 0\\ \\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in R-\{0;-1\}\\ \\ (1)/(x)\cdot 3x(2x+2) + (1)/(2x+2)\cdot 3x(2x+2)= (2)/(3)\cdot 3x(2x+2)\\ \\3(2x+2)+3x=2x(2x+2)\\ \\6x+6+3x=4x^2+4x\\ \\-4x^2+9x-4x+6=0\\ \\-4x^2+5x+6=0\ \ \ \Rightarrow\ \ \Delta=5^2-4\cdot(-4)\cdot6=25+96=121\\ \\`


`x_1= (-5- √(121) )/(2\cdot (-4))= (-5-11)/(-8) = (-16)/(-8) =2\ \ \ is\ integer\\ \\x_2= (-5+ √(121) )/(2\cdot (-4))= (-5+11)/(-8) = (6)/(-8) =- (3)/(4) \ \ \ is\ not\ the\ integer`