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  4. Total number of values of 'a' , so that x2-x-a=0 has integral roots, where a∈ N and 6 le a le 100 , is equal to

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Q. Total number of values of $ 'a' $ , so that $ {{x}^{2}}-x-a=0 $ has integral roots, where $ a\in N $ and $ 6\le a\le 100 $ , is equal to

Jharkhand CECEJharkhand CECE 2013

Solution:

$x^{2}-x-a=0, D=1+4 a=$ odd $D$ must be perfect square of some odd integer.
Let $D=(2 \lambda+1)^{2} \Rightarrow 1+4 a=1+4 \lambda^{2}+4 \lambda \Rightarrow a=\lambda(\lambda+1)$,
as $a \in[6,100] \Rightarrow a=6,12,20,30,42,56,72,90$
Thus, ' $a^{\prime}$ can attain 8 different values.