Question

if the equation x³+3px²+3qx+r=0 and x²+2px+q=0 have a common root then 4(p²-q)(q²-pr)=

Answer 1

dear student


x3+3px2+3qx+r=0 and x2+2px+q=0let common root be aa3+3pa2+3qa+r=0 and a2+2pa+q=0multiplying equation 2 by a and subtracting we getpa2+2qa+r=0 alsoa2+2pa+q=0 multiplying second equation by p and subtracting we ga=r−pq2(p2−q)we have a2+2pa+q=0putting value of a, we get(r−pq2(p2−q))2+2p(r−pq2(p2−q))+q=0(r−pq)2=−4p(r−pq)(p2−q)−4q(p2−q)2=−4(p2−q)(pr−p2q+p2q−q2)=4(p2−q)(q2−pr)so4(p2−q)(q2−pr)=(r−pq)2
regards