Question

Giải phương trình và hệ phương trình

a)x⁴-9x² +20=0

b) |x+1| +|x-1|=1+|x² -1|

c) \(\sqrt{x^2+x+25}-\sqrt{x^2+x+9}=2\)

d)\(\begin{cases} x+|y|=3\\ 2x-|y|=3 \end{cases} \)

Answer 1

c.

Đặt \(\sqrt{x^2+x+9}=t>0\Rightarrow x^2+x=t^2-9\)

Pt trở thành:

\(\sqrt{t^2-9+25}-t=2\)

\(\Leftrightarrow\sqrt{t^2+16}=t+2\)

\(\Leftrightarrow t^2+16=t^2+4t+4\)

\(\Leftrightarrow t=3\)

\(\Leftrightarrow\sqrt{x^2+x+9}=3\)

\(\Leftrightarrow x^2+x=0\)

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

Answer 2

a.

\(x^4-9x^2+20=0\)

\(\Leftrightarrow x^4-4x^2-5x^2+20=0\)

\(\Leftrightarrow x^2\left(x^2-4\right)-5\left(x^2-4\right)=0\)

\(\Leftrightarrow\left(x^2-4\right)\left(x^2-5\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2=4\\x^2=5\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\pm2\\x=\pm\sqrt{5}\end{matrix}\right.\)

Answer 3

b.

\(\left|x+1\right|+\left|x-1\right|=1+\left|x-1\right|.\left|x+1\right|\)

\(\Leftrightarrow\left|x-1\right|.\left|x+1\right|-\left|x+1\right|-\left|x-1\right|+1=0\)

\(\Leftrightarrow\left|x+1\right|\left(\left|x-1\right|-1\right)-\left(\left|x-1\right|-1\right)=0\)

\(\Leftrightarrow\left(\left|x+1\right|-1\right)\left(\left|x-1\right|-1\right)=0\)

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

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

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

Answer 4

d.

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

\(\Leftrightarrow\left\{{}\begin{matrix}3x=6\\\left|y\right|=3-x\end{matrix}\right.\)

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

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