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Question

For the equation 4x2+x+4x2+1+x2+1x2+x+1=316
Which of the following statement(s) is/are correct?

A
The equation has 4 real and distinct roots.
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B
The equation has 3 real and distinct roots.
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C
The sum of all real and distinct roots is 2.
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D
The sum of all real and distinct roots is 1.
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Solution

The correct options are
B The equation has 3 real and distinct roots.
C The sum of all real and distinct roots is 2.
4x2+x+4x2+1+x2+1x2+x+1=3164+xx2+1+11+xx2+1=316
Assuming xx2+1=t
4+t+11+t=316(t+4)(t+1)+1(t+1)=3166(t2+5t+5)=31(t+1)6t2t1=0(3t+1)(2t1)=0t=13,12

When t=13
xx2+1=13x2+3x+1=0D=94=5>0
Two real and distinct roots.

When t=12
xx2+1=12x22x+1=0(x1)2=0
Two real and equal roots.

Hence, the given equation has 3 real and distinct roots, and
The sum of distinct roots
=3+1=2

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