Question

3/ Giải phương trình bằng cách đặt ẩn phụ:

1. x²-|x|-2=0

2. x²+2x+|x+1|-5=0

3. x²+2x-5|x+1|+5=0

4. x²-2x+5|x-1|+5=0

5. x²-4x+2|x-2|+1=0

6. 4x²-20x+4|2x-5|+13=0

Answer 1

a/ Đặt \(\left|x\right|=t\ge0\Rightarrow t^2-t-2=0\Rightarrow\left[{}\begin{matrix}t=-1\left(l\right)\\t=2\end{matrix}\right.\)

\(\Rightarrow\left|x\right|=2\Rightarrow x=\pm2\)

b/ \(\Leftrightarrow\left(x+1\right)^2+\left|x+1\right|-6=0\)

Đặt \(\left|x+1\right|=t\ge0\Rightarrow t^2+t-6=0\Rightarrow\left[{}\begin{matrix}t=-3\left(l\right)\\t=2\end{matrix}\right.\)

\(\Rightarrow\left|x+1\right|=2\Rightarrow\left[{}\begin{matrix}x+1=2\\x+1=-2\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=1\\x=-3\end{matrix}\right.\)

c/ \(\Leftrightarrow\left(x+1\right)^2-5\left|x+1\right|+4=0\)

Đặt \(\left|x+1\right|=t\ge0\Rightarrow t^2-5t+4=0\Rightarrow\left[{}\begin{matrix}t=1\\t=4\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}\left|x+1\right|=1\\\left|x+1\right|=4\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x+1=1\\x+1=-1\\x+1=4\\x+1=-4\end{matrix}\right.\)

Answer 2

d. \(\Leftrightarrow\left(x-1\right)^2+5\left|x-1\right|+4=0\)

Đặt \(\left|x+1\right|=t\ge0\Rightarrow t^2+5t+4=0\Rightarrow\left[{}\begin{matrix}t=-1\left(l\right)\\t=-4\left(l\right)\end{matrix}\right.\)

Vậy pt vô nghiệm

e. \(\Leftrightarrow\left(x-2\right)^2+2\left|x-2\right|-3=0\)

Đặt \(\left|x-2\right|=t\ge0\)

\(\Rightarrow t^2+2t-3=0\Rightarrow\left[{}\begin{matrix}t=1\\t=-3\left(l\right)\end{matrix}\right.\)

\(\Rightarrow\left|x-2\right|=1\Leftrightarrow\left[{}\begin{matrix}x-2=1\\x-2=-1\end{matrix}\right.\)

f. \(\Leftrightarrow\left(2x-5\right)^2+4\left|2x-5\right|-12=0\)

Đặt \(\left|2x-5\right|=t\ge0\)

\(\Rightarrow t^2+4t-12=0\Rightarrow\left[{}\begin{matrix}t=2\\t=-6\left(l\right)\end{matrix}\right.\)

\(\Rightarrow\left|2x-5\right|=2\Rightarrow\left[{}\begin{matrix}2x-5=2\\2x-5=-2\end{matrix}\right.\)