Consider the equation x2 + 4x - n = 0, where n [20, 100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to
(1) 6
The roots are x = -4 16 + 4n2 = -2 4 + n, so they are integers ⇔ 4 + n is a perfect square.
∵ 20 n 100, 24 4 + n 104, so
4 + n = 25, 36, 49, 64, 81, 100
⇒ n = 21, 32, 45, 60, 77, 96
∴ Number of values of n = 6
Answer: (1)
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