a) \(3x^2+5x-6=0\)
\(\Rightarrow\left\{{}\begin{matrix}S=x_1+x_2=-\dfrac{5}{3}\\P=x_1.x_2=-2\end{matrix}\right.\)
\(A=\left(3x_1-2x_2\right)\left(3x_2-2x_1\right)\)
\(\Leftrightarrow A=9x_1x_2-6x^2_1-6x^2_2+4x_1x_2\)
\(\Leftrightarrow A=13x_1x_2-6\left(x^2_1+x^2_2\right)\)
\(\Leftrightarrow A=13x_1x_2-6\left[\left(x_1+x_2\right)^2-2x_1x_2\right]\)
\(\Leftrightarrow A=13x_1x_2+12x_1x_2-6\left(x_1+x_2\right)^2\)
\(\Leftrightarrow A=25x_1x_2-6\left(x_1+x_2\right)^2\)
\(\Leftrightarrow A=25.\left(-2\right)-6.\left(-\dfrac{5}{3}\right)^2\)
\(\Leftrightarrow A=-50-\dfrac{50}{3}\)
\(\Leftrightarrow A=-\dfrac{190}{3}\)
b) \(B=\dfrac{x_2}{x_1-1}+\dfrac{x_1}{x_2-1}\)
\(\Leftrightarrow B=\dfrac{x_2\left(x_2-1\right)+x_1\left(x_1-1\right)}{\left(x_1-1\right)\left(x_2-1\right)}\)
\(\Leftrightarrow B=\dfrac{x^2_2-x_2+x^2_1-x_1}{x_1x_2-x_1-x_2+1}\)
\(\Leftrightarrow B=\dfrac{\left(x_1+x_2\right)^2-2x_1x_2-\left(x_1+x_2\right)}{x_1x_2-\left(x_1+x_2\right)+1}\)
\(\Leftrightarrow B=\dfrac{\left(-\dfrac{5}{3}\right)^2-2.\left(-2\right)+\dfrac{5}{3}}{-2+\dfrac{5}{3}+1}\)
\(\Leftrightarrow B=\dfrac{\dfrac{25}{9}+\dfrac{5}{3}+4}{\dfrac{2}{3}}=\dfrac{\dfrac{76}{9}}{\dfrac{2}{3}}=\dfrac{38}{3}\)
