Question

`x^2 -x-1=0` có 2 nghiệm phân biệt x1, x2. Không giải pt, tính \(A=\sqrt{x_1^4-x_1^2+x_2-1}-\left|x_2\right|\)

Answer 1

Theo hệ thức Viet: \(\left\{{}\begin{matrix}x_1+x_2=1\\x_1x_2=-1\end{matrix}\right.\)

Do \(x_1\) là nghiệm nên: \(x_1^2-x_1-1=0\Rightarrow\left\{{}\begin{matrix}x_1^2=x_1+1\\x_1^2-1=x_1\end{matrix}\right.\)

\(\Rightarrow x_1^4-x_1^2=x_1^2\left(x_1^2-1\right)=x_1\left(x_1+1\right)=x_1^2+x_1=2x_1+1\)

\(\Rightarrow A=\sqrt{2x_1+x_2}-\left|x_2\right|=\sqrt{x_1+x_2+x_1}-\left|x_2\right|=\sqrt{x_1+1}-\left|x_2\right|\)

\(=\sqrt{x_1^2}-\left|x_2\right|=\left|x_1\right|-\left|x_2\right|\)

\(\Rightarrow A^2=x_1^2+x_2^2-2\left|x_1x_2\right|=\left(x_1+x_2\right)^2-2x_1x_2-2\left|x_1x_2\right|\)

\(=1+2-2=1\)

\(\Rightarrow A=\pm1\)