Ta có: \(x^4-8x^3+21x^2-24x+9=0\)

\(\Leftrightarrow x^4-5x^3+3x^2-3x^3+15x^2-9x+3x^2-5x+9=0\)

\(\Leftrightarrow\left(x^2-5x+3\right)\left(x^2-3x+3\right)=0\)

\(\Leftrightarrow x^2-5x+3=0\)

\(\text{Δ}=\left(-5\right)^2-4\cdot1\cdot3=25-12=13\)

Vì Δ>0 nên phương trình có hai nghiệm phân biệt là:

\(\left\{{}\begin{matrix}x_1=\dfrac{5-\sqrt{13}}{2}\\x_2=\dfrac{5+\sqrt{13}}{2}\end{matrix}\right.\)

\(sin^23x-cos^24x=sin^25x-cos^26x\)

\(\Leftrightarrow2sin^23x-2cos^24x=2sin^25x-2cos^26x\)

\(\Leftrightarrow2sin^23x-1+1-2cos^24x=2sin^25x-1+1-2cos^26x\)

\(\Leftrightarrow-cos6x-cos8x=-cos10x-cos12x\)

\(\Leftrightarrow cos6x-cos12x+cos8x-cos10x=0\)

\(\Leftrightarrow sin9x.sin6x+sin9x.sin4x=0\)

\(\Leftrightarrow sin9x.\left(sin6x+sin4x\right)=0\)

\(\Leftrightarrow2sin9x.sin5x.cosx=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin9x=0\\sin5x=0\\cosx=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{9}\\x=\dfrac{k\pi}{5}\\x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)

ĐK: \(\hept{\begin{cases}x\ge2\\y\ge-\frac{1}{3}\end{cases}}\)

\(\sqrt{x-2}+x^3-6x^2+12x=\sqrt{3y+1}+27y^3+27y^2+9y+9\)

<=> \(\sqrt{x-2}+x^3-6x^2+12x-8=\sqrt{3y+1}+27y^3+27y^2+9y+1\)

<=> \(\sqrt{x-2}+\left(x-2\right)^3=\sqrt{3y+1}+\left(3y+1\right)^3\)

<=> \(\left(\sqrt{x-2}-\sqrt{3y+1}\right)+\left[\left(x-2\right)^3-\left(3y+1\right)^3\right]=0\)

<=> \(\frac{x-3y-3}{\sqrt{x-2}+\sqrt{3y+1}}+\left(x-3y-3\right)\left[\left(x-2\right)^2+\left(x-2\right)\left(3y+1\right)+\left(3y+1\right)^2\right]=0\)

<=> \(\left(x-3y-3\right)\left(\frac{1}{\sqrt{x-2}+\sqrt{3y+1}}+\left(x-2\right)^2+\left(x-2\right)\left(3y+1\right)+\left(3y+1\right)^2\right)=0\)

<=> \(x-3y-3=0\)

vì \(\frac{1}{\sqrt{x-2}+\sqrt{3y+1}}+\left(x-2\right)^2+\left(x-2\right)\left(3y+1\right)+\left(3y+1\right)^2>0\)

<=> x = 3y + 3

Thế vào phương trình trên ta có: 

\(2+2\left(3y+3\right)^2-2y^2+3\left(3y+3\right)y-4\left(3y+3\right)-3y=0\)

<=> \(25y^2+30y+8=0\Leftrightarrow\orbr{\begin{cases}y=-\frac{2}{5}\\y=-\frac{4}{5}\end{cases}}\)không thỏa mãn đk 

Vậy hệ vô nghiệm.

\(PT\Leftrightarrow-5x^2-24x+60=\left(x^2+5x-10\right)^2\\ \Leftrightarrow-5x^2-24x+60=x^4+10x^3+5x^2-100x+100\\ \Leftrightarrow x^4+10x^3+10x^2-76x+40=0\\ \Leftrightarrow x^4+4x^3-10x^2+6x^3+24x^2-60x-4x^2-16x+40=0\\ \Leftrightarrow\left(x^2+4x-10\right)\left(x^2+6x-4\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x^2+4x-10=0\\x^2+6x-4=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-2+\sqrt{14}\\x=-2-\sqrt{14}\\x=-3+\sqrt{13}\\x=-3-\sqrt{13}\end{matrix}\right.\)

Bài giải

 Cộng cả 2 vế với 4x^2+4 
>>x^4+4x^2+4=4x^2+24x+36 
>>(x^2+2)^2=4(x+3)^2 
>>x^2+2=2(x+3)(1) hoặc x^2+2=-2(x+3)(2) 
(1)>>x^2-2x-4=0>>x=1(+-)căn 5 
(2)>>x^2+2x+8=0(vô nghiệm)

~Hok tốt~

Cộng cả 2 vế với 4x^2+4 
>>x^4+4x^2+4=4x^2+24x+36 
>>(x^2+2)^2=4(x+3)^2 
>>x^2+2=2(x+3)(1) hoặc x^2+2=-2(x+3)(2) 
(1)>>x^2-2x-4=0>>x=1(+-)căn 5 
(2)>>x^2+2x+8=0(vô nghiệm

ctfy,uggvvvvvvujhmtfgg

ĐKXĐ: \(x\geq -2\).

Nhận thấy x = -2 không là nghiệm của pt.

Xét x khác -2.

\(PT\Leftrightarrow\sqrt[3]{x^3+8}-\left(2x+4\right)=\dfrac{24x-18}{x^2-2x-7}-6\)

\(\Leftrightarrow\dfrac{\left(x+2\right)\left(x^2-6x-4\right)}{\sqrt[3]{x^3+8}+x+2}=\dfrac{-6\left(x^2-6x-4\right)}{x^2-2x-7}\)

\(\Leftrightarrow\dfrac{x+2}{\sqrt[3]{x^3+8}+x+2}=\dfrac{-6}{x^2-2x-7}\left(1\right)\) hoặc x² - 6x - 4 = 0.

\(\left(1\right)\Rightarrow\left(x+2\right)\left(x^2-2x-1\right)=-6\sqrt[3]{x^3+8}\)

+) Nếu x \(\geq 7\) thì \(\left(x+2\right)\left(x^2-2x-1\right)>0\ge-6\sqrt{x^3+8}\) (loại)

+) Nếu \(x\le7\) thì \(\left(x+2\right)\left(x^2-2x-1\right)\ge-2\left(x+2\right)>-6\sqrt[3]{3\left(x+2\right)}\ge-6\sqrt[3]{x^3+8}\) (loại)

Do đó (1) vô nghiệm.

Do đó \(x^2-6x-4=0\Leftrightarrow\left[{}\begin{matrix}x=3+\sqrt{13}\left(TMĐK\right)\\x=3-\sqrt{13}\left(loại\right)\end{matrix}\right.\)

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