Multiple choice

Consider the equation ${x^2} + 2x - n = 0$, where $n \in N$ and $n \in \left[ {5,100} \right]$. Total number of different values of 'n' so that the given equation has integral roots, is

  1. 4

  2. 6

  3. 8

  4. 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For x^2 + 2x - n = 0 to have integral roots, the discriminant D = 4 + 4n must be a perfect square, say k^2. So 4(1+n) = k^2, meaning 1+n must be a perfect square. For n in [5, 100], 1+n is in [6, 101]. Perfect squares in this range are 9, 16, 25, 36, 49, 64, 81, 100. There are 8 such values.

AI explanation

For the equation x^2 + 2x - n = 0 to have integer roots, the discriminant (2^2 - 4(1)(-n)) must be a perfect square. This simplifies to 4 + 4n = 4(1 + n), meaning 1 + n must be a perfect square. The values of n in the range 5 to 100 that make 1 + n a perfect square are 8, 15, 24, 35, 48, 63, 80, and 99, giving a total of 8 values.

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