Question

Cho PT \(ax^2-bx+b=0\)(ab>0) có các nghiệm là x₁, x₂. CMR x₁>0. x₂>0 và \(\sqrt{\frac{x_1}{x_2}}+\sqrt{\frac{x_2}{x_1}}-\sqrt{\frac{b}{a}}=0\)

Answer 1

Theo Viet: \(\left\{{}\begin{matrix}x_1+x_2=\frac{b}{a}=\frac{ab}{a^2}>0\\x_1x_2=\frac{b}{a}=\frac{ab}{a^2}>0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}x_1>0\\x_2>0\end{matrix}\right.\)

\(\sqrt{\frac{x_1}{x_2}}+\sqrt{\frac{x_2}{x_1}}-\sqrt{\frac{b}{a}}=\frac{x_1+x_2}{\sqrt{x_1x_2}}-\sqrt{\frac{b}{a}}=\frac{\frac{b}{a}}{\sqrt{\frac{b}{a}}}-\sqrt{\frac{b}{a}}=\sqrt{\frac{b}{a}}-\sqrt{\frac{b}{a}}=0\)