If a + b + c = 0, then a root of the equation Δ=a-xcbcbxbac-x=0 is

Applying C1→C1+C2+C3, we get Δ=a+b+c-xcba+b+c-xb-xaa+b+c-xac-x=-xcb-xb-xa-xac-x  [∵a+b+c=0]∆clearly equals 0 when x = 0. It is unnecessary to evaluate the determinant further.