if the equations x^3+3Px^2+3qx+r=0 and x^2+2Px+q=0 have a common root then 4(p^2-q)(q^2-Pr)= ?

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 getr-pq2(p2-q)2+2pr-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

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