If α,β  be the roots of x2+px+q=0  and α+h,β+h  are roots of x2+rx+s=0 , then :

α+β=−p,αβ=q                   α+β+2h=−r ,         (α+h)(β+h)=s                         −p+2h=−r ⇒                  h=p−r2 Now,          αβ+h(α+β)+h2=s ⇒      q+(p−r2)(−p)+(p−r2)2=s ⇒     q−(p2−pr)2+p2+r2−2pr4=s ⇒     4q−2p2+2pr+p2+r2−2pr=4s ⇒                 4q−p2+r2−4s=0 ⇒                       r2−4s=p2−4q