Question

[Đỗ Quỳnh Anh]

Giúp em chi tiết câu này với

Answer 1

[Hồng Phúc]

ĐK: \(x\ge0;x\ne\dfrac{1}{4}\)

\(A=\dfrac{4\sqrt{x}+3}{2\sqrt{x}-1}\in Z\)

\(\Leftrightarrow\dfrac{2\left(2\sqrt{x}-1\right)+5}{2\sqrt{x}-1}\in Z\)

\(\Leftrightarrow2+\dfrac{5}{2\sqrt{x}-1}\in Z\)

\(\Leftrightarrow2\sqrt{x}-1\inƯ_5=\left\{\pm1;\pm5\right\}\)

\(\Leftrightarrow2\sqrt{x}\in\left\{0;2;6\right\}\)

\(\Leftrightarrow\sqrt{x}\in\left\{0;1;3\right\}\)

\(\Leftrightarrow x\in\left\{0;1;9\right\}\)

Answer 2

[Nguyễn Lê Phước Thịnh]

Để A là số nguyên thì \(4\sqrt{x}+3⋮2\sqrt{x}-1\)

\(\Leftrightarrow5⋮2\sqrt{x}-1\)

\(\Leftrightarrow2\sqrt{x}-1\in\left\{-1;1;5\right\}\)

\(\Leftrightarrow2\sqrt{x}\in\left\{0;2;6\right\}\)

hay \(x\in\left\{0;1;9\right\}\)