Question

# If (2x2−3x+1)(2x2+5x+1)=9x2, then the absolute value of the sum of all real roots of the equation is

Solution

## (2x2−3x+1)(2x2+5x+1)=9x2 As x=0 is not a root of the equation, dividing the equation by x2, (2x+1x−3)(2x+1x+5)=9 Assuming 2x+1x=y, (y−3)(y+5)=9⇒y2+2y−24=0⇒(y+6)(y−4)=0⇒y=−6,4 Case 1:  When y=−6 2x+1x=−6⇒2x2+6x+1=0D=36−8=28>0 Sum of real roots =−62=−3 Case 2:  When y=4 2x+1x=4⇒2x2−4x+1=0D=16−8=8>0 Sum of real roots =2 Hence, the absolute value of the sum of all real roots is 1.

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