Question

Q= \(\dfrac{3-\sqrt{x}}{5\sqrt{x}+7}+\dfrac{3\sqrt{x}+4}{3\sqrt{x}-2}-\dfrac{42\sqrt{x}+34}{15x+11\sqrt{x}-14}\)

a, Rút gọn

b, Tìm x sao cho Q ≤-\(\dfrac{2}{3}\)

c, Tìm các giá trị nguyên của x sao cho Q nhận giá trị nguyên

Answer 1

a: \(Q=\dfrac{9\sqrt{x}-6-3x+2\sqrt{x}+15x+41\sqrt{x}+28-42\sqrt{x}-34}{\left(5\sqrt{x}+7\right)\left(3\sqrt{x}-2\right)}\)

\(=\dfrac{12x+10\sqrt{x}-12}{\left(5\sqrt{x}+7\right)\left(3\sqrt{x}-2\right)}\)

\(=\dfrac{12x+18\sqrt{x}-8\sqrt{x}-12}{\left(3\sqrt{x}-2\right)\left(5\sqrt{x}+7\right)}\)

\(=\dfrac{6\sqrt{x}\left(2\sqrt{x}+3\right)-4\left(2\sqrt{x}+3\right)}{\left(3\sqrt{x}-2\right)\left(5\sqrt{x}+7\right)}\)

\(=\dfrac{4\sqrt{x}+6}{5\sqrt{x}+7}\)

c: Để Q là số nguyên thì \(20\sqrt{x}+30⋮5\sqrt{x}+7\)

\(\Leftrightarrow5\sqrt{x}+7\inƯ\left(2\right)\)

hay \(x\in\varnothing\)