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Question

if~~α and ~~β are the roots of the equation x^2 -p (x+1) -q = 0 then value of
(α^2+2α+1)/α^2+2α+q) +(β^2+2β+1)/(β^2+2 β+q) is ?
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Solution

Dear student
Given: α and β are the roots of x2-px+1-q=0i.e. x2-px-p-q=0=x2-px-p+q=0α+β=-coeff. of xcoeff. of x2=--p1=p ...1αβ=constant termcoeff. of x2=-p+q1=-p+qαβ=-p-qq=-p-αβq=-α+β-αβ ...2 using 1To find the value of :α2+2α+1α2+2α+q+β2+2β+1β2+2β+q=α+12α2+2α-α-β-αβ+β+12β2+2β-α-β-αβ using 2=α+12α2+α-β-αβ+β+12β2+β-α-αβ=α+12α-βα+1+β+12β-αβ+1=α+1α-β-β+1α-β=α+1-β-1α-β=α-βα-β=1Note:a±b2=a2+b2±2ab
Regards

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