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

Question

Consider the equation $ x^{2}+2 x-n=0 $, where $ n \in N $ and $ n \in[5,100] . $ The total number of different values of $ n $ so that the given equation has integral roots is, $ 8 $, $ 3 $, $ 6 $, $ 4 $

MARKS App · Accepted Answer

We have, x 2 + 2 x - n = 0 We know that for a Quadratic equation a x 2 + b x + c = 0 , a ≠ 0 the roots are x = - b ± b 2 - 4 a c 2 a . ⇒ x = - 1 ± n + 1 Thus, n + 1 should be a perfect square for integral roots. Now, n ∈ 5 , 100 ⇒ n + 1 ∈ 6 , 101 Perfect square values of n + 1 are 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 Hence, number of values is 8 .

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