An algebra problem by akash more

$(x -a)(x-b) + k$ has roots c and d , therefore $(x -a)(x-b) - k= (x-c)(x-d)$

This implies $(x-c)(x-d) + k = (x -a)(x-b)$ ; which is quadratic equation with roots a and b

$(x-a)(x-b)\quad -\quad k\quad =\quad 0\quad \\ \\ observe\quad the\quad equations,\\ \\ as\quad c\quad and\quad d\quad are\quad the\quad roots\quad of\quad the\quad equation\\ \\ we\quad have\quad cd=\quad ab-k\\ and\quad c+d\quad =\quad a+b\\ \\ so\quad we\quad have\quad \\ \\ ab=\quad cd+k\\ and\quad a+b=\quad c+d\\ \\ which\quad clearly\quad is\quad the\quad relations\quad between\quad the\quad roots\quad of\quad the\quad equation\\ \\ { x }^{ 2 }-(c+d)x+cd+k\quad =\quad (x-c)(x-d)\quad +\quad k\quad$