For real value of x , the range of x^2+2 x+1 x^2+2 x-1 is

Step-by-step solution

aligned & Let y= x^2+2 x+1 x^2+2 x-1 & y (x^2+2 x-1 )=x^2+2 x+1 & y x^2+2 x y-y=x^2+2 x+1 & y x^2-x^2+2 x y-2 x-y-1=0 & (y-1) x^2+2(y-1) x-y-1=0 aligned For real values of x, b^2-4 a c 0 aligned & [2(y-1)]^2+4(y-1)(y+1) 0 & 4(y-1)^2+4(y-1)(y+1) 0 & 4(y-1)(y-1+y+1) 0 & 4(y-1)(2 y) 0 & 8 y(y-1) 0 & At y=0, x^2+2 x+1=0 & (x+1)^2=0 x=-1 R & At y=1, x^2+2 x+1=x^2+2 x-1 & 1 -1 y 1 & y (- , 0] (1, ) aligned