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If (x+a)(x+1991) + 1 Can Be Factored As (x+b)(x+c) Where B,C Are Integers Then Find The Sum Of All Possible Integral Values Of A?

Sujoy Das

07/02/2018 2 0

Answer:

Method 1:

x^2+(1991+a)x+1991a+1

Now for integers d should be either 0 or perfect square so,

(1991+a)^2-4*1991a-4

(1991-a)^2-4

(1989-a)(1993-a)

They both can be equal so for d = 0

a = 1989, a = 1993

Method 2:

b and c are roots

(-b+a) (-b+1991) +1 = 0

(-b+a) (-b+1991) = -1*1

a-b = -1 and 1991 - b = 1

a-b=1 and 1991-b = -1

2 values of a and b

a = 1989 and 1993 Sum = 3982.

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