a, với x tm ĐKXĐ
\(A=\dfrac{\sqrt{x}}{\sqrt{x}+4}+\dfrac{3\sqrt{x}-1}{\sqrt{x}-4}+\dfrac{3x+11\sqrt{x}-4}{16-x}\)
\(\Leftrightarrow A=\dfrac{\sqrt{x}}{\sqrt{x}+4}+\dfrac{3\sqrt{x}-1}{\sqrt{x}-4}-\dfrac{3x+11\sqrt{x}-4}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+4\right)}\)
\(\Leftrightarrow A=\dfrac{\sqrt{x}\left(\sqrt{x}-4\right)}{\left(\sqrt{x}+4\right)\left(\sqrt{x}-4\right)}+\dfrac{\left(3\sqrt{x}-1\right)\left(\sqrt{x}+\text{4}\right)}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+4\right)}-\dfrac{3x+11\sqrt{x}-4}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+4\right)}\)
\(\Leftrightarrow A=\dfrac{x-4\sqrt{x}+3x+12\sqrt{x}-\sqrt{x}-4-3x-11\sqrt{x}+4}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+4\right)}\)
\(\Leftrightarrow A=\dfrac{x-4\sqrt{x}}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+4\right)}=\dfrac{\sqrt{x}\left(\sqrt{x}-4\right)}{\left(\sqrt{x}+4\right)\left(\sqrt{x}-4\right)}=\dfrac{\sqrt{x}}{\sqrt{x}+4}\)
b, với x tm ĐKXĐ
\(A\le\dfrac{1}{2}\Leftrightarrow\dfrac{\sqrt{x}}{\sqrt{x}+4+4}\le\dfrac{1}{2}\Leftrightarrow\dfrac{\sqrt{x}}{\sqrt{x}+4}-\dfrac{1}{2}\le0\)
\(\Leftrightarrow\dfrac{2\sqrt{x}}{2\left(\sqrt{x}+4\right)}-\dfrac{\sqrt{x}+4}{2\left(\sqrt{x}+4\right)}\le0\)
\(\Leftrightarrow\dfrac{\sqrt{x}-4}{2\left(\sqrt{x}+4\right)}\le0\Leftrightarrow\sqrt{x}-4\le0\left(2\left(\sqrt{x}+4\right)>0\forall x\ge0\right)\)
\(\Leftrightarrow\sqrt{x}\le4\Leftrightarrow x\le16\)
Kết hợp điều kiện xác định \(\Rightarrow0\le x< 16\)
c, với x tm đkxđ
\(A=\dfrac{\sqrt{x}}{\sqrt{x}+4}\Rightarrow P=\dfrac{4\sqrt{x}}{\sqrt{x}+4}-1\)
\(\Leftrightarrow P=\dfrac{4\sqrt{x}-\sqrt{x}-4}{\sqrt{x}+4}=\dfrac{3\sqrt{x}-4}{\sqrt{x}+4}=3-\dfrac{16}{\sqrt{x}+4}\)
\(P\in Z\Leftrightarrow\dfrac{16}{\sqrt{x}+4}\in Z\)
Ta có x có giá trị nguyên thì \(\sqrt{x}\) có thể có giá trị nguyên hoặc là số vô tỉ
\(+\sqrt{x}\in I\Leftrightarrow\sqrt{x}+\text{4}\in I\Leftrightarrow\dfrac{16}{\sqrt{x}+4}\notin Z\left(loại\right)\)
\(+\sqrt{x}\in Z\Leftrightarrow\sqrt{x}+4\in Z\Leftrightarrow\dfrac{16}{\sqrt{x}+4}\in Z\)
\(\Leftrightarrow\sqrt{x}+4\inƯ\left(16\right)=\left\{\pm1;\pm2;\pm4;\pm8;\pm16\right\}\)
\(\Leftrightarrow\sqrt{x}\in\left\{-5;-3;-6;-2;-8;0;-12;4;-20;12\right\}\)
\(\Leftrightarrow\sqrt{x}\in\left\{0;4;12\right\}\)
\(\Leftrightarrow x\in\left\{0;16;144\right\}\)
kết hợp đkxđ
\(x\in\left\{0;144\right\}\)
