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Question
If every pair of the equtions x2+px+qr=0,x2+qr+rp=0,x2+rx+pq=0 have a none-zero common root, then the sum of three common roots
If every pair of the equtions x2+px+qr=0,x2+qr+rp=0,x2+rx+pq=0 have a none-zero common root, then the sum of three common roots
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Solution
The correct option is B −(p+q+r)2
suppose(α,β) be the roots of x2+px+qr=0
(β,γ) are the roots of x2+qr+rp=0
and (α,γ) are the root of x2+rx+pq=0
α+β=−pβ+γ=−qα+γ=−r
adding all roots equations we get
2α+2β+2γ=−p−q−r2(α+β+γ)=−(p+q+r)α+β+γ=−12(p+q+r)
then the sum of three common root is −(p+q+r)2
A is the correct answer
suppose
(α,β) be the roots of x2+px+qr=0
(β,γ) are the roots of x2+qr+rp=0
and (α,γ) are the root of x2+rx+pq=0
α+β=−pβ+γ=−qα+γ=−r
adding all roots equations we get
2α+2β+2γ=−p−q−r2(α+β+γ)=−(p+q+r)α+β+γ=−12(p+q+r)
then the sum of three common root is −(p+q+r)2
A is the correct answer
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