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Question

If (2x23x+1)(2x2+5x+1)=9x2, then the absolute value of the sum of all real roots of the equation is 


Solution

(2x23x+1)(2x2+5x+1)=9x2
As x=0 is not a root of the equation, dividing the equation by x2,
(2x+1x3)(2x+1x+5)=9
Assuming 2x+1x=y,
(y3)(y+5)=9y2+2y24=0(y+6)(y4)=0y=6,4

Case 1:  When y=6
2x+1x=62x2+6x+1=0D=368=28>0
Sum of real roots =62=3
Case 2:  When y=4
2x+1x=42x24x+1=0D=168=8>0
Sum of real roots =2

Hence, the absolute value of the sum of all real roots is 1.

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