Question

[Hoang Duc Thinh]

Giai phuong trinh

a.\(\left(x+1\right)^3+\left(x-2\right)^3=\left(2x-1\right)^3\)

b.\(x^4=24x+32\)

c.\(\left(x-6\right)^4+\left(x-8\right)^4=16\)

Answer 1

[ST]

a, \(\Leftrightarrow\left(x+1+x-2\right)\left[\left(x+1\right)^2-\left(x+1\right)\left(x-2\right)+\left(x-2\right)^2\right]-\left(2x-1\right)^3=0\)

\(\Leftrightarrow\left(2x-1\right)\left(x^2+2x+1-x^2+x+2+x^2-4x+4\right)-\left(2x-1\right)^3=0\)

\(\Leftrightarrow\left(2x-1\right)\left(x^2-x+7-\left(2x-1\right)^2\right)=0\)

\(\Leftrightarrow\left(2x-1\right)\left(x^2-x+7-4x^2+4x-1\right)=0\)

\(\Leftrightarrow\left(2x-1\right)\left(-3x^2+3x+6\right)=0\)

\(\Leftrightarrow-3\left(2x-1\right)\left(x^2-x-2\right)=0\)

\(\Leftrightarrow\left(2x-1\right)\left(x+1\right)\left(x-2\right)=0\)

=>x=1/2 hoặc x=-1 hoặc x=2

Vậy pt có tập nghiệm là S={1/2;-1;2}

b, \(x^4=24x+32\Leftrightarrow x^4-24x-32=0\)

\(\Leftrightarrow x^4-2x^3-4x^2+2x^3-4x^2-8x+8x^2-16x-32=0\)

\(\Leftrightarrow x^2\left(x^2-2x-4\right)+2x\left(x^2-2x-4\right)+8\left(x^2-2x-4\right)=0\)

\(\Leftrightarrow\left(x^2-2x-4\right)\left(x^2+2x+8\right)=0\)

\(\Leftrightarrow x^2-2x-4=0\) (vì x^2+2x+8 > 0)

\(\Leftrightarrow\left(x-1\right)^2-5=0\Leftrightarrow\left(x-1\right)^2=5\Leftrightarrow x-1=\pm\sqrt{5}\Leftrightarrow x=1\pm\sqrt{5}\)

Vậy...

c, \(\left(x-6\right)^4+\left(x-8\right)^4=16\)

Đặt x-6=t => x-8=t-2

Ta có: \(t^4+\left(t-2\right)^4=16\Leftrightarrow t^4+t^4-8t^3+24t^2-32t+16=16\)

\(\Leftrightarrow2t^4-8t^3+24t^2-32t=0\Leftrightarrow t^4-4t^3+12t^2-16t=0\)

\(\Leftrightarrow t^4-2t^3-2t^3+4t^2+8t^2-16t=0\)

\(\Leftrightarrow t^3\left(t-2\right)-2t^2\left(t-2\right)+8t\left(t-2\right)=0\)

\(\Leftrightarrow\left(t-2\right)\left(t^3-2t^2+8t\right)=0\Leftrightarrow\left(t-2\right)t\left(t^2-2t+8\right)=0\)

Mà t^2-2t+8=(t-1)^2+7 > 0

\(\Rightarrow\orbr{\begin{cases}t-2=0\\t=0\end{cases}\Leftrightarrow\orbr{\begin{cases}x-6-2=0\\x-6=0\end{cases}\Leftrightarrow}\orbr{\begin{cases}x=8\\x=6\end{cases}}}\)

Vậy...