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=316⇒4+xx2+1+11+xx2+1=316 Assuming xx2+1=t ⇒4+t+11+t=316⇒(t+4)(t+1)+1(t+1)=316⇒6(t2+5t+5)=31(t+1)⇒6t2−t−1=0⇒(3t+1)(2t−1)=0⇒t=−13,12 When t=−13 xx2+1=−13⇒x2+3x+1=0D=9−4=5>0 Two real and distinct roots. When t=12 xx2+1=12⇒x2−2x+1=0⇒(x−1)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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