a: \(\sqrt{\left(x-3\right)^2}=3-x\)
=>|x-3|=3-x
=>x-3<=0
=>x<=3
b:
ĐKXĐ: x>=1
\(\sqrt{x+2\sqrt{x-1}}=2\)
=>\(\sqrt{x-1+2\cdot\sqrt{x-1}+1}=2\)
=>\(\sqrt{\left(\sqrt{x-1}+1\right)^2}=2\)
=>\(\sqrt{x-1}+1=2\)
=>\(\sqrt{x-1}=1\)
=>x-1=1
=>x=2(nhận)
c:
ĐKXĐ: x>=1
\(\sqrt{x-2\sqrt{x-1}}=\sqrt{x-1}-1\)
=>\(\sqrt{x-1-2\cdot\sqrt{x-1}+1}=\sqrt{x-1}-1\)
=>\(\sqrt{\left(\sqrt{x-1}-1\right)^2}=\sqrt{x-1}-1\)
=>\(\left|\sqrt{x-1}-1\right|=\sqrt{x-1}-1\)
=>\(\sqrt{x-1}-1\ge0\)
=>\(\sqrt{x-1}\ge1\)
=>x-1>=1
=>x>=2
d: ĐKXĐ: x>=\(\sqrt{\frac32}\)
\(\sqrt{2x^2-3}=\sqrt{4x-3}\)
=>\(2x^2-3=4x-3\)
=>\(2x^2-4x=0\)
=>2x(x-2)=0
=>x(x-2)=0
=>x=0(loại) hoặc x=2(nhận)
e: ĐKXĐ: x>=5/3
\(\sqrt{x^2-x}=\sqrt{3x-5}\)
=>\(x^2-x=3x-5\)
=>\(x^2-4x+5=0\)
=>\(\left(x-2\right)^2+1=0\) (vô lý)
=>x∈∅
f: \(\sqrt{x^4-2x^2+1}=x-1\)
=>\(\sqrt{\left(x^2-1\right)^2}=x-1\)
=>\(\left|x^2-1\right|=x-1\)
=>\(\begin{cases}x-1\ge0\\ \left(x^2-1\right)^2=\left(x-1\right)^2\end{cases}\Rightarrow\begin{cases}x\ge1\\ \left(x^2-1-x+1\right)\left(x^2-1+x-1\right)=0\end{cases}\)
=>\(\begin{cases}x\ge1\\ \left(x^2+x-2\right)\left(x^2-x\right)=0\end{cases}\Rightarrow\begin{cases}x\ge1\\ \left(x+2\right)\cdot\left(x-1\right)\cdot x\cdot\left(x-1\right)=0\end{cases}\Rightarrow x=1\)
g: |3x+1|=|x+1|
=>\(\left[\begin{array}{l}3x+1=x+1\\ 3x+1=-x-1\end{array}\right.\Rightarrow\left[\begin{array}{l}2x=0\\ 4x=-2\end{array}\right.\Rightarrow\left[\begin{array}{l}x=0\\ x=-\frac12\end{array}\right.\)
h: \(\left|x^2-3\right|=\left|x-\sqrt3\right|\)
=>\(\left|\left(x-\sqrt3\right)\left(x+\sqrt3\right)\right|=\left|x-\sqrt3\right|\)
=>\(\left|x-\sqrt3\right|\left(\left|x+\sqrt3\right|-1\right)=0\)
TH1: \(\left|x-\sqrt3\right|=0\)
=>\(x-\sqrt3=0\)
=>\(x=\sqrt3\)
TH2: \(\left|x+\sqrt3\right|-1=0\)
=>\(\left|x+\sqrt3\right|=1\)
=>\(\left[\begin{array}{l}x+\sqrt3=1\\ x+\sqrt3=-1\end{array}\right.\Rightarrow\left[\begin{array}{l}x=1-\sqrt3\\ x=-1-\sqrt3\end{array}\right.\)
i: \(\left|x^2-1\right|+\left|x+1\right|=0\)
=>\(\left|x+1\right|\left(\left|x-1\right|+1\right)=0\)
=>|x+1|=0
=>x+1=0
=>x=-1
k: ĐKXĐ: x>=2 hoặc x<=-2
\(\sqrt{x^2-4}+\sqrt{x^2+4x+4}=0\)
=>\(\sqrt{\left(x-2\right)\left(x+2\right)}+\sqrt{\left(x+2\right)^2}=0\)
=>\(\sqrt{x+2}\left(\sqrt{x-2}+\sqrt{x+2}\right)=0\)
=>\(\sqrt{x+2}=0\)
=>x+2=0
=>x=-2(nhận)
