Toán Lớp 9: Tìm x để P=(\sqrt[x]-3)/(\sqrt[x]+2) có giá trị là số nguyên
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Toán Lớp 9: Tìm x để P=(\sqrt[x]-3)/(\sqrt[x]+2) có giá trị là số nguyên
Comments ( 1 )
\quad P = \dfrac{\sqrt x – 3}{\sqrt x + 2}\qquad (x\geqslant 0)\\
\to P = \dfrac{\sqrt x + 2 – 5}{\sqrt x + 2}\\
\to P = 1 – \dfrac{5}{\sqrt x +2}\\
\text{Khi đó:}\\
P\in \Bbb Z\Leftrightarrow \dfrac{5}{\sqrt x + 2}\in\Bbb Z\\
\Leftrightarrow \sqrt x + 2 = \dfrac{5}{k}\quad (k\in\Bbb Z)\\
\text{Ta có:}\\
\sqrt x \geqslant 0,\quad \forall x\geqslant 0\\
\to \sqrt x + 2\geqslant 2\\
\text{Do đó:}\\
\quad \dfrac5k \geqslant 2\\
\to 0 < k \leqslant \dfrac52\\
\to k\in\{1;2\}\\
\to \sqrt x + 2 \in \left\{5;\dfrac52\right\}\\
\to \sqrt x \in \left\{3;\dfrac12\right\}\\
\to x \in \left\{9;\dfrac14\right\}
\end{array}\)