a, \(ĐKXĐ:\left\{{}\begin{matrix}x\ne1\\x\ne-1\end{matrix}\right.\)

b, \(R=\left(\frac{\left(x-1\right)^2}{3x+\left(x-1\right)^2}-\frac{1-2x^2+4x}{x^3-1}+\frac{1}{x-1}\right):\frac{x^2+x}{x^3+x}\)

\(=\left(\frac{x^2-2x+1}{x^2+x+1}-\frac{1-2x^2+4x}{\left(x-1\right)\left(x^2+x+1\right)}+\frac{1}{x-1}\right):\frac{x\left(x+1\right)}{x\left(x^2+1\right)}\)

\(=\left(\frac{\left(x^2-2x+1\right)\left(x-1\right)-1+2x^2-4x+x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+1}{x+1}\right)\)

\(=\frac{x^3-3x^2+3x-1+3x^2-3x}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+1}{x+1}\)

\(=\frac{x^3-1}{x^3-1}.\frac{x^2+1}{x+1}=\frac{x^2+1}{x+1}\)

\(b,\) Để R = 0

\(\Leftrightarrow\frac{x^2+1}{x+1}=0\Leftrightarrow x^2+1=0\) ( vô lý)

Vậy ko có giá trị nào của x để R =0

\(c,\left|R\right|=1\Leftrightarrow\left[{}\begin{matrix}R=-1\\R=1\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\frac{x^2+1}{x+1}=-1\\\frac{x^2+1}{x+1}=1\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x^2+1=-x-1\\x^2+1=x+1\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2+x+2=0\\x^2-x=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\)

a) A= \(\frac{x\left(1-x^2\right)^2}{1+x^2}\): \(\left\{\left[\frac{\left(1-x\right)\left(1+x+x^2\right)}{1-x}+x\right]\left[\frac{\left(1+x\right)\left(1-x+x^2\right)}{1+x}-x\right]\right\}\)
A= \(\frac{x\left(1-x^2\right)^2}{1+x^2}\): (1+x+x²+x)(1-x+x²-x)
A=\(\frac{x\left(1-x^2\right)^2}{1+x^2}\): (1+2x+x²)(1-2x+x²)
A= \(\frac{x\left(1-x^2\right)^2}{1+x^2}\): (1+x)²(1-x)²
A= \(\frac{x\left(1-x^2\right)^2}{1+x^2}\): (1+x)(1+x)(1-x)(1-x)
A= \(\frac{x\left(1-x^2\right)\left(1-x^2\right)}{1+x^2}.\frac{1}{\left(1-x^2\right)\left(1-x^2\right)}\)
A= \(\frac{x}{1+x^2}\)
b)Thay x= \(-\frac{1}{2}\) vào biểu thức A, có:
A= \(\frac{\frac{-1}{2}}{1+\left(\frac{-1}{2}\right)^2}\)
\(\Leftrightarrow\)A= \(\frac{-2}{5}\)
Vậy A= \(\frac{-2}{5}\) khi x=\(-\frac{1}{2}\)
c) Để 2A=1 thì \(\frac{2x}{1+x^2}\)=1
\(\Leftrightarrow\)\(\frac{2x}{1+x^2}\)-1=0
\(\Leftrightarrow\)2x-1-x²=0
\(\Leftrightarrow\)-(2x+1+x²)=0
\(\Leftrightarrow\)x²-2x+1=0
\(\Leftrightarrow\)(x-1)²=0
\(\Leftrightarrow\)x-1=0
\(\Leftrightarrow\)x=1
Vậy x=1 thì 2A=1

a) ĐKXĐ \(\sqrt{x}\ne2\Leftrightarrow x\ne4\)

b) \(A=\frac{\sqrt{x}+2}{\sqrt{x}+3}-\frac{5}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}-\frac{1}{\sqrt{x}-2}\)

\(A=\frac{x-4-5-\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}=\frac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-4\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}\)

\(A=\frac{\sqrt{x}-4}{\sqrt{x}-2}\)

c) x = 6 + \(4\sqrt{2}\) = \(\left(2+\sqrt{2}\right)^2\)

=> A = \(\frac{\sqrt{\left(2+\sqrt{2}\right)^2}-4}{\sqrt{\left(2+\sqrt{2}\right)^2}-2}=\frac{\sqrt{2}-2}{\sqrt{2}}\)

2.

a) đkxđ: \(x\ne4;x\ne9\)

A=\(\left(\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-3\right)+\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)-\left(\sqrt{x}+2\right)\left(x-4\right)}{\left(x-4\right)\left(\sqrt{x}-3\right)}\right).\frac{\sqrt{x}+1}{\sqrt{x}+2}\)

=\(\left(\frac{x-\sqrt{x}-6+x\sqrt{x}-9\sqrt{x}-2x+18-x\sqrt{x}+2x-4\sqrt{x}-8}{\left(x-4\right)\left(\sqrt{x}-3\right)}\right).\frac{\sqrt{x}+1}{\sqrt{x}+2}\)

\(\frac{x-14\sqrt{x}-5}{\left(x-4\right)\left(\sqrt{x}-3\right)}.\frac{\sqrt{x}+1}{\sqrt{x}+2}\)

b) A = -2/5

(k biết là do đề sai hay mình sai chứ đến đây nản quá! bạn làm nốt nhé!)

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