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  • If a,b , c belong to R and the equation ax^2+bx+c=0 and x^3+3x^2+3x+2=0 have two common roots , then

If a,b , c belong to R and the equation ax^2+bx+c=0 and x^3+3x^2+3x+2=0 have two common roots , then

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Solution:     x^3+3x^2+3x+2=0

             Rightarrow         (x+1)^3=-1    Rightarrow     (x+1)=-1,-omega ,-omega ^2

              Rightarrow         x=-2,-1-omega ,-1-omega ^2

             As a,b,c Belong to R the roots of ax^2+bx+c=0 are both real

          or both real  or imaginary .

         herefore    roots of ax^2+bx+c=0 must be -1-omega ,-1-omega ^2.

          Thus ,         -1-omega -1-omega ^2=-b/a

        and       (-1-omega )(-1-omega ^2)=c/a       

          Rightarrow       1=b/a hspace0.2cm,1=c/a     Rightarrow     a=b=c

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Deependra Verma

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