The correct option is B rational and equal roots
f(x)=(x−12)2+2(x−12)(x+12)+(x+12)2
Roots are decreased by 12, so the transformed equation is,
f(x+12)=0
⇒x2+2x(x+24)+(x+24)2=0⇒x2+2x2+48x+x2+48x+242=0⇒4x2+96x+242=0⇒x2+24x+122=0⇒(x+12)2=0⇒x=−12,−12
Hence, roots are rational and equal.