Question

[....]

Giải các phương trình sau:

a \(\left(X^2+2x\right)^2-3\left(x^2+2x\right)+2=0\)

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

c \(x^4-4x^3+x+3=0\)

d \(x^4-2x^3+x=2\)

Answer 1

[Nguyễn Hoàng Minh]

\(a,\) Đặt \(x^2+2x=a\), pt trở thành:

\(a^2-3a+2=0\\ \Leftrightarrow\left[{}\begin{matrix}a=1\\a=2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x^2+2x-1=0\left(1\right)\\x^2+2x-2=0\left(2\right)\end{matrix}\right.\)

\(\left[{}\begin{matrix}\Delta\left(1\right)=4+4=8\\\Delta\left(2\right)=4+8=12\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left[{}\begin{matrix}x=\dfrac{-2-\sqrt{8}}{2}\\x=\dfrac{-2+\sqrt{8}}{2}\end{matrix}\right.\\\left[{}\begin{matrix}x=\dfrac{-2-\sqrt{12}}{2}\\x=\dfrac{-2+\sqrt{12}}{2}\end{matrix}\right.\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=-1-\sqrt{2}\\x=-1+\sqrt{2}\\x=-1-\sqrt{3}\\x=-1+\sqrt{3}\end{matrix}\right.\)

\(b,\) Đặt \(x^2+x=b\), pt trở thành:

\(b\left(b+1\right)-6=0\\ \Leftrightarrow b^2+b-6=0\\ \Leftrightarrow\left[{}\begin{matrix}b=2\\b=-3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x^2+x-2=0\\x^2+x+3=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}\left[{}\begin{matrix}x=1\\x=-2\end{matrix}\right.\\x\in\varnothing\left[x^2+x+3=\left(x+\dfrac{1}{2}\right)^2+\dfrac{11}{4}>0\right]\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=1\\x=-2\end{matrix}\right.\)

\(d,x^4-2x^3+x=2\\ \Leftrightarrow x^4-2x^3+x-2=0\\\Leftrightarrow\left(x^3+1\right)\left(x-2\right)=0 \\ \Leftrightarrow\left(x+1\right)\left(x-2\right)\left(x^2+x+1\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=-1\\x=2\\x^2+x+1=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=2\\x\in\varnothing\left[x^2+x+1=\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\right]\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=-1\\x=2\end{matrix}\right.\)

Answer 2

[Akai Haruma]

Lời giải:

a. 

PT $\Leftrightarrow (x^2+2x)^2-(x^2+2x)-2[(x^2+2x)-1]=0$

$\Leftrightarrow (x^2+2x)(x^2+2x-1)-2(x^2+2x-1)=0$

$\Leftrightarrow (x^2+2x-1)(x^2+2x-2)=0$

$\Leftrightarrow x^2+2x-1=0$ hoặc $x^2+2x-2=0$

$\Leftrightarrow x=-1\pm \sqrt{2}$ hoặc $x=-1\pm \sqrt{3}$

b.

PT $\Leftrightarrow (x^2+x)^2+(x^2+x)-6=0$

$\Leftrightarrow (x^2+x)^2-2(x^2+x)+3(x^2+x)-6=0$

$\Leftrightarrow (x^2+x)(x^2+x-2)+3(x^2+x-2)=0$

$\Leftrightarrow (x^2+x-2)(x^2+x+3)=0$

$\Leftrightarrow x^2+x-2=0$ (chọn) hoặc $x^2+x+3=0$ (loại do $x^2+x+3=(x+0,5)^2+2,75>0$)

$\Leftrightarrow x=-1\pm \sqrt{3}$

c. Nghiệm khá xấu. Bạn coi lại đề.

d.

PT $\Leftrightarrow x^3(x-2)+(x-2)=0$

$\Leftrightarrow (x^3+1)(x-2)=0$

$\Leftrightarrow x^3+1=0$ hoặc $x-2=0$

$\Leftrightarrow x=-1$ hoặc $x=2$