If a,b,c are in Geometric Progression, then prove that the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common rootIf da,cb,fc are in an Arithmetic Progressionb=acax2+2acx+c=0(ax+c)(ax+c)=0x=−casince both roots are roots are equal, both equations have the same roots. Thereforeda=eb=fcWhy is this contradiction arising?

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2sze1c1se3nh · Accepted Answer

This contradiction is arising because of the wrong statement that &#039;since both roots are roots are equal, both equations have the same roots&#039;. Both roots of the first equation are same, does not imply that every quadratic equation having that root as one of the roots, has both roots equal.Now proving the claim:We wish to show that x=−ca is a root of the equation dx2+2ex+f=0 if da,cb and fc are in AP. First let r=cb and m=eb−da=fc−eb. Then applying quadratic formula in dx2+2ex+f=0, we get,x=−e+e2−dfd=−br+b2r2−ac(r−m)(r+m)a(r−m)=−br+bma(r−m)=−caHence proved

If a,b,c are in Geometric Progression, then prove that the equations ax^2+2bx+c=0 and dx^2+

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