Q.

If the equation x4-4x3+ax2+bx+1=0 has four positive roots, find the value of a+b=

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answer is 2.

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Detailed Solution

Let x1, x2, x3, x4 are the roots of the equation

x4-4x3+ax2+bx+1=0

x1+x2+x3+x4=4 and x1x2x3x4=1

  AM=x1+x2+x3+x44=44=1

and GM=x1x2x3x41/4=(1)1/4=1

i.e., AM=GM

which is true only when x1=x2=x3=x4=1

Hence, given equation has all roots identical, equal to 1 i.e., equation have form

(x-1)4=0

  x4-4x3+6x2-4x+1=0

On comparing Eqs. (i) and (ii), we get

 a=6, b=-4 

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