Quartic: x^4 +ax^3 +bx^2 +cx+d=0 (x^2 +px+h)(x^2 +qx+k)=0 p+q=a h+k=b−pq=b−m (pq=m say) pk+qh=c hk=d q(h+k)=q(b−m) p(h+k)=p(b−m) ph+qk=c k(p−q)=p(b−m)−c h(p−q)=c−q(b−m) multiplying d(a^2 −4m)= ac(b−m)−m(b−m)^2 −c^2 m(b−m)^2 +(ac−4d)m +a(ad−bc)+c^2 =0 −−−−−−−−−−−−−−− m^3 −2bm^2 +(b^2 +ac−4d)m +a(ad−bc)+c^2 =0 −−−−−−−−−−−−−−− p , q=(a/2)±(√((a^2 /4)−m)) −−−−−−−−−−−−−−− h=((c−q(b−m))/(p−q)) k=((p(b−m)−c)/(p−q)) −−−−−−−−−−−−−−− x=−(p/2)±(√((p^2 /4)−{((c−q(b−m))/(p−q))})) or −−−−−−−−−−−−−− x=−(q/2)±(√((q^2 /4)−{((p(b−m)−c)/(p−q))})) −−−−−−−−−−−−−−−