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Question

If the product of two of the roots x48x3+21x220x+5=0 is 5, then the roots are

A
2±52,5±52
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B
3±52,5±52
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C
5,1±2
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D
4±52,5±52
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Solution

The correct option is A 3±52,5±52
Let α,β,γ,δ be the roots of the equation x48x3+21x220x+5=0,
where α×β=5
Now, S1=α+β+γ+δ=8 ...(1)
S2=αβ+αγ+αδ+βγ+βδ+γδ=21
αβ+(α+β)(γ+δ)+γδ=21 ...(2)
S3=αβγ+αβδ+αγδ+βγδ=20
αβ(γ+δ)+γδ(α+β)=20
5(γ+δ)+γδ(α+β)=20 ...(3)
S4=αβγδ=5
5γδ=5γδ=1 ...(4)
Substituting this in (3), we get
5(γ+δ)+(α+β)=20 ...(5)
Now solving (1) and (5), we get
4(γ+δ)=12γ+δ=3
As (γδ)2=(γ+δ)24γδ
=94=5
γδ=±5
Hence, from the given options we can conclude that γ,δ=3±52 and α,β=5±52

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