. Find all integer values of a such that the quadratic expression
(x + a)(x + 1991) + 1
can be factored as a product (x + b)(x + c) where b and c are integers.
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adthedaddy
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manihar.sidharth
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Here's something I got after 20 mins of wasted effort and after that copying it from google
Equate x coefficients
b + c = 1991+a
bc = 1991a+1
Eliminate a
1991(b + c) - bc = 3964080
Let c = b + k, then
b² + (k-3982)b - 1991k + 3964080 = 0
with solutions
b = -k/2 + 1991+ (1/2)√(k²+4),
b = -k/2 +1991 - (1/2)√(k²+4)
Since √(k²+4) is an integer only if k = 0, the only two solutions are
b = c = 1992, a = 1993
b = c = 1990, a = 1989
Answer: 2
Equate x coefficients
b + c = 1991+a
bc = 1991a+1
Eliminate a
1991(b + c) - bc = 3964080
Let c = b + k, then
b² + (k-3982)b - 1991k + 3964080 = 0
with solutions
b = -k/2 + 1991+ (1/2)√(k²+4),
b = -k/2 +1991 - (1/2)√(k²+4)
Since √(k²+4) is an integer only if k = 0, the only two solutions are
b = c = 1992, a = 1993
b = c = 1990, a = 1989
Answer: 2
