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Question

If the sum of two roots of x42x3+4x2+6x21=0 is zero, then which of the following is/are true?

A
one of the roots of the equation is 1+i6
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B
all roots of the equation are real
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C
the equation has only two real roots
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D
sum of all the real roots of the equation is 0
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Solution

The correct option is D sum of all the real roots of the equation is 0
x42x3+4x2+6x21=0
Let α,β,γ and δ be the roots.
Then S1=α+β+γ+δ=2
S2=αβ+αγ+αδ+βγ+βδ+γδ=4
S3=αβγ+αβδ+βγδ+αγδ=6
S4=αβγδ=21

Let α+β=0 (1)
γ+δ=2 (2)

αβ+αγ+αδ+βγ+βδ+γδ=4
αβ+αγ+αδαγαδ+γδ=4 [β=α]
αβ+γδ=4 (3)

αβγ+αβδ+βγδ+αγδ=6
αβγ+αβδαγδ+αγδ=6 [β=α]
αβ(γ+δ)=6
αβ=3 (4)

αβγδ=21
From (4),γδ=7 (5)

Solving (1) and (4), we get
α=3, β=3 or α=3, β=3

Solving (2) and (5), we get
γ=1+i6, δ=1i6 or γ=1i6, δ=1+i6

Roots are 3,3,1+i6,1i6

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