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Điều kiện để biểu thức P tồn tại là: \(\left\{{}\begin{matrix}x\ne4\\x>0\end{matrix}\right.\)
P = \(\left(\frac{4\sqrt{x}}{2-\sqrt{x}}-\frac{8x}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\right):\left(\frac{\sqrt{x}\left(\sqrt{x}-4\right)+x+2\sqrt{x}}{\sqrt{x}\left(x+2\sqrt{x}\right)}\right)\)
= \(\left(\frac{4\sqrt{x}\left(2+\sqrt{x}\right)-8x}{4-x}\right):\left(\frac{x-4\sqrt{x}+x+2\sqrt{x}}{\sqrt{x}\left(x+2\sqrt{x}\right)}\right)\)
= \(\frac{8\sqrt{x}-4x}{4-x}\cdot\frac{\sqrt{x}\left(x+2\sqrt{x}\right)}{2x-2\sqrt{x}}\)
= \(\frac{4\sqrt{x}\left(2-\sqrt{x}\right)}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\cdot\frac{x\left(\sqrt{x}+2\right)}{2\left(x-\sqrt{x}\right)}\)
=\(\frac{2x\sqrt{x}}{x-\sqrt{x}}\)
a, Với \(x\ge0;x\ne\frac{16}{9};4\)
\(P=\frac{2\sqrt{x}-4}{3\sqrt{x}-4}-\frac{4+2\sqrt{x}}{\sqrt{x}-2}+\frac{x+13\sqrt{x}-20}{3x-10\sqrt{x}+8}\)
\(=\frac{2x-8\sqrt{x}+8-4\sqrt{x}-6x+16+x+13\sqrt{x}-20}{\left(3\sqrt{x}-4\right)\left(\sqrt{x}-2\right)}\)
\(=\frac{-3x+\sqrt{x}+4}{\left(3\sqrt{x}-4\right)\left(\sqrt{x}-2\right)}=\frac{-\left(3\sqrt{x}-4\right)\left(\sqrt{x}+1\right)}{\left(3\sqrt{x}-4\right)\left(\sqrt{x}-2\right)}=\frac{\sqrt{x}+1}{2-\sqrt{x}}\)
b, \(P\ge-\frac{3}{4}\Rightarrow\frac{\sqrt{x}+1}{2-\sqrt{x}}+\frac{3}{4}\ge0\Leftrightarrow\frac{4\sqrt{x}+4+6-3\sqrt{x}}{8-4\sqrt{x}}\ge0\Leftrightarrow\frac{\sqrt{x}+10}{8-4\sqrt{x}}\ge0\)
\(\Rightarrow2-\sqrt{x}\ge0\Leftrightarrow x\le4\)Kết hợp với đk vậy \(0\le x< 4\)
\(=\left(\frac{\sqrt{x}\left(\sqrt{2}+2\right)+\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{2}+2\right)}\right).\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{4\text{x}}}\)
\(=\left(\frac{\sqrt{2\text{x}}+2\sqrt{x}+x-2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{2}+2\right)}\right).\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{4\text{x}}}\)
\(=\frac{\sqrt{2\text{x}}+x}{\left(\sqrt{x}-2\right)\left(\sqrt{2}+2\right)}.\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{4\text{x}}}\)
\(=\frac{\sqrt{2\text{x}}+x}{\sqrt{2}+2}.\frac{\sqrt{x}-2}{\sqrt{4\text{x}}}\)
\(=\frac{x\sqrt{2}-2\sqrt{2\text{x}}+x\sqrt{x}-2\text{x}}{2\sqrt{2\text{x}}+4\sqrt{x}}\)
tick cho mình nha
a) \(ĐKXĐ:\hept{\begin{cases}x\ge0\\x\ne1\\x\ne16\end{cases}}\)
\(B=\frac{2\left(x+4\right)}{x-3\sqrt{x}-4}+\frac{\sqrt{x}}{\sqrt{x}+1}-\frac{8}{\sqrt{x}-4}\)
\(\Leftrightarrow B=\frac{2x+8+\sqrt{x}\left(\sqrt{x}-4\right)-8\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-4\right)}\)
\(\Leftrightarrow B=\frac{2x+8+x-4\sqrt{x}-8\sqrt{x}-8}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-4\right)}\)
\(\Leftrightarrow B=\frac{3x-12\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-4\right)}\)
\(\Leftrightarrow B=\frac{3\sqrt{x}}{\sqrt{x}+1}\)
b) Để B nguyên'
\(\Leftrightarrow3\sqrt{x}⋮\sqrt{x}+1\)
\(\Leftrightarrow3\left(\sqrt{x}+1\right)-3⋮\sqrt{x}+1\)
\(\Leftrightarrow3⋮\sqrt{x}+1\)
\(\Leftrightarrow\sqrt{x}+1\inƯ\left(3\right)=\left\{\pm1;\pm3\right\}\)
\(\Leftrightarrow\sqrt{x}\in\left\{0;2\right\}\)(Đã loại những giá trị âm)
\(\Leftrightarrow x\in\left\{0;4\right\}\)
Vậy để \(B\inℤ\Leftrightarrow x\in\left\{0;2\right\}\)