Question-112058

Question Number 112058 by mohssinee last updated on 05/Sep/20

Answered by MJS_new last updated on 06/Sep/20

\Rightarrow x = - \frac{19 \pm \sqrt{4 a^{2} - 407}}{2}

4 a^{2} - 407 = b^{2} with b \in Z

4 a^{2} = b^{2} + 407

let 4 a^{2} = {(b + n)}^{2} with n \in Z

\Rightarrow {(b + n)}^{2} - b^{2} = 407

\Rightarrow b = \frac{407 - n^{2}}{2 n}

∣ \frac{407 - n^{2}}{2 n} ∣⩾ 1 \Rightarrow - 19 ⩽ n ⩽ 19

\Rightarrow

n = \pm 11 \land b = \pm 13 \Rightarrow a = \pm 12 \land x = {\begin{cases} - 3 \\ - 16 \end{cases}

n = \pm 1 \land b = \pm 203 \Rightarrow a = \pm 102 \land x = {\begin{cases} - 111 \\ 92 \end{cases}

no other solutions

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