Question

1/ \(\sqrt{x-2}\) - \(\sqrt{1-3x}\) = 0
2/ \(\sqrt{15-x}\) + \(\sqrt{3-x}\) = 6
3/ \(\sqrt{x-1}\) - \(\sqrt{x+1}\) = 2
4/ \(\sqrt{\frac{1}{2}-3x}\) + \(\sqrt{3-\frac{1}{3}x}\) = 0

5/ \(2\sqrt{x-5}\) = \(\sqrt{-3x-2}\)

6/ \(\sqrt{x-1}\) - \(\sqrt{5x-1}\) = \(\sqrt{3x-2}\)

Answer 1

1/ \(\sqrt{x-2}-\sqrt{1-3x}=0\\ đk:\left\{{}\begin{matrix}x-2\ge0\\1-3x\ge0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ge2\\x\le\frac{1}{3}\end{matrix}\right.\)

=> pt vô no

2/ \(\sqrt{15-x}+\sqrt{3-x}=6\\ đk\left\{{}\begin{matrix}15-x\ge0\\3-x\ge0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\le15\\x\le3\end{matrix}\right.\Leftrightarrow x\le3\)

\(pt\Leftrightarrow15-x+3-x+2\sqrt{\left(15-x\right)\left(3-x\right)}=36\)

\(\Leftrightarrow2\sqrt{\left(15-x\right)\left(3-x\right)}=2x+36\)

\(\Leftrightarrow4\left(15-x\right)\left(3-x\right)=\left(2x+18\right)^2\left(đk:x\ge-9\right)\)

\(\Leftrightarrow-144x=144\Leftrightarrow x=-1\left(nhan\right)\)

Answer 2

Câu 1: ĐKXĐ: \(\left\{{}\begin{matrix}x-2\ge0\\1-3x\ge0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x\ge2\\x\le\frac{1}{3}\end{matrix}\right.\)

\(\Rightarrow\) Không tồn tại x thỏa mãn ĐKXĐ \(\Rightarrow\) pt vô nghiệm

Câu 2:

ĐKXĐ: \(x\le3\)

\(\Leftrightarrow15-x+3-x+2\sqrt{\left(15-x\right)\left(3-x\right)}=36\)

\(\Leftrightarrow x+9=\sqrt{x^2-18x+45}\) (\(x\ge-9\))

\(\Leftrightarrow x^2+18x+81=x^2-18x+45\)

\(\Leftrightarrow36x=-36\Rightarrow x=-1\)

Câu 3:

ĐKXĐ: \(x\ge1\)

\(\Leftrightarrow\sqrt{x-1}=2+\sqrt{x+1}\)

\(\Leftrightarrow x-1=4+x+1+4\sqrt{x+1}\)

\(\Leftrightarrow\sqrt{x+1}=-\frac{3}{2}\)

Phương trình vô nghiệm

Answer 3

Câu 4:

ĐKXĐ: \(\left\{{}\begin{matrix}\frac{1}{2}-3x\ge0\\3-\frac{1}{3}x\ge0\end{matrix}\right.\) \(\Rightarrow x\le\frac{1}{6}\)

Do \(\left\{{}\begin{matrix}\sqrt{\frac{1}{2}-3x}\ge0\\\sqrt{3-\frac{1}{3}x}\ge0\end{matrix}\right.\) nên đẳng thức xảy ra khi và chỉ khi:

\(\left\{{}\begin{matrix}\frac{1}{2}-3x=0\\3-\frac{1}{3}x=0\end{matrix}\right.\) \(\Rightarrow\) phương trình vô nghiệm

Câu 5:

ĐKXĐ: \(\left\{{}\begin{matrix}x-5\ge0\\-3x-2\ge0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x\ge5\\x\le-\frac{2}{3}\end{matrix}\right.\)

\(\Rightarrow\) Không tồn tại x thỏa mãn ĐKXĐ \(\Rightarrow\) pt vô nghiệm

Câu 6:

ĐKXĐ: \(x\ge1\)

\(\Leftrightarrow\sqrt{x-1}=\sqrt{3x-2}+\sqrt{5x-1}\)

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

\(\Leftrightarrow2-7x=2\sqrt{\left(3x-2\right)\left(5x-1\right)}\)

Do \(x\ge1\Rightarrow2-7x< 0\Rightarrow VT< 0\)

Mà \(VP=2\sqrt{\left(3x-2\right)\left(5x-1\right)}>0\Rightarrow VP>VT\)

Phương trình vô nghiệm