Question

[Truong Cong Vinh]

giải phương trình tìm x

(x-5)⁴ + (x-2)⁴ = 17

ai nhanh nhất cho 1 like

Answer 1

[Nguyễn Huy Tú]

Ta có \(\left(x-5\right)^4+\left(x-2\right)^4=1^4+2^4=2^4+1^4\)

TH1 \(\left\{{}\begin{matrix}\left(x-5\right)^4=1^4\\\left(x-2\right)^4=2^4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x-5=1\\x-5=-1\end{matrix}\right.\\\left[{}\begin{matrix}x-2=2\\x-2=-2\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x=6\\x=4\end{matrix}\right.\\\left[{}\begin{matrix}x=4\\x=0\end{matrix}\right.\end{matrix}\right.\)

TH2 \(\left\{{}\begin{matrix}\left(x-5\right)^4=2^4\\\left(x-2\right)^4=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x-5=2\\x-5=-2\end{matrix}\right.\\\left[{}\begin{matrix}x-2=1\\x-2=-1\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x=7\\x=3\end{matrix}\right.\\\left[{}\begin{matrix}x=3\\x=1\end{matrix}\right.\end{matrix}\right.\)

Answer 2

[TAO CHƠI FREE FIRE]

Đặt $x-\frac{7}{2}=a$. Khi đó PT trở thành:

$(a-\frac{3}{2}{)}^4+(a+\frac{3}{2}{)}^4=17$

$\iff 2a^4+27a^2+\frac{81}{8}=17$

$\iff 2a^4+27a^2=\frac{55}{8}$

$\iff a^4+\frac{27}{2}{a}^2=\frac{55}{16}$

$\iff (a^2+\frac{27}{4}{)}^2=49$

$\Rightarrow [\begin{array}{c}{a}^2+\frac{27}{4}=7\\ a^2+\frac{27}{4}=-7<0\left(\text{vô lý}\right)\end{array}$

$\Rightarrow a^2=\frac{1}{4}\Rightarrow a=\pm \frac{1}{2}$

$\Rightarrow x=a+\frac{7}{2}=[\begin{array}{c}4\\ 3\end{array}$