Question

Tìm x \(\in Z\), sao cho :(x²-1)(x²-4)(x²-7)(x²-10)<0

Answer 1

\(\left(x^2-1\right)\left(x^2-4\right)\left(x^2-7\right)\left(x^2-10\right)< 0\)

\(\Leftrightarrow\left[\left(x^2-1\right)\left(x^2-10\right)\right].\left[\left(x^2-4\right)\left(x^2-7\right)\right]< 0\)

\(\Leftrightarrow\left(x^4-11x^2+10\right)\left(x^4-11x^2+28\right)< 0\)

\(\Leftrightarrow x^4-11x^2+10,x^2-11x^2+28\) là 2 số trái dấu

Mà \(x^4-11x^2+10< x^4-11x^2+28\)

\(\Leftrightarrow\left\{{}\begin{matrix}x^4-11x^2+10< 0\\x^4-11x^2+28>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x^2-\dfrac{11}{2}\right)^2-\dfrac{81}{4}< 0\\\left(x^2-\dfrac{11}{2}\right)^2-\dfrac{9}{4}>0\end{matrix}\right.\)

\(\Leftrightarrow\dfrac{9}{4}< \left(x^2-\dfrac{11}{2}\right)^2< \dfrac{81}{4}\)

\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{3}{2}< x^2-\dfrac{11}{2}< \dfrac{9}{2}\\-\dfrac{3}{2}>x^2-\dfrac{11}{2}>-\dfrac{9}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}7< x^2< 10\\4>x^2>1\end{matrix}\right.\)

Vì \(x\in Z\Leftrightarrow x^2\in Z\Leftrightarrow x^2=9\Leftrightarrow\left[{}\begin{matrix}x=3\\x=-3\end{matrix}\right.\)

Vậy....