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The Integral values of a for which the equation (a+2)X^2 +2(a+1)X +a = 0 will have both roots integers (1) 0(2) - 1(3) - 2(4) - 3

The Integral values of a for which the equation (a+2)X^2 +2(a+1)X +a = 0 will have both roots integers (1) 0(2) - 1(3) - 2(4) - 3

Grade:12th pass

1 Answers

Snehita
11 Points
6 years ago
Given (a+2)X^2 +2(a+1)X +a = 0 has integral roots for integral values of aLet p=(a+2) q=2(a+1) r=a=>[-q±√(q^2 - 4pr)]/2p ={Z}Condition 1:√(q^2 - 4pr) ={Z}=> √(4(a+1)^2 - 4(a+2)a) ={Z}=> √(4a^2 + 4 + 8a - 4a^2 - 8a) ={Z}=> √4 ={Z}=> ±2 ={Z}Condition 1 is satisfied for any value of aCondition 2:(-q ± 2)/2p ={Z}=> (-2(a+1) ± 2)/2(a+2) ={Z}=> (-a-1±1)/a+2 ={Z}Case 1:(-a-1-1)/a+2 ={Z}=> -(a+2)/(a+2) ={Z}=> -1 ={Z}Case 1 of Condition 1 is true for all values of aCase 2:(-a-1+1)/(a+2) ={Z}=>-a/(a+2) ={Z}=> X = {a: [-a/(a+2)] = z & z belongs to Z }Values of a satisfying the above condition are 0,-1,-3,-4.[Note : The notation `={Z}` implies `belongs to the set of integers`.]

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