\(9x^2-9x+1=0\)
\(x_1+x_2=-\dfrac{b}{a}=1;x_1x_2=\dfrac{c}{a}=\dfrac{1}{9}\)
\(A=x_1\sqrt{x_2}+x_2\sqrt{x_1}\)
\(=\sqrt{x_1x_2}\left(\sqrt{x_1}+\sqrt{x_2}\right)=\sqrt{\dfrac{1}{9}}\cdot\left(\sqrt{x_1}+\sqrt{x_2}\right)\)
\(=\dfrac{1}{3}\cdot\left(\sqrt{x_1}+\sqrt{x_2}\right)\)
\(\left(\sqrt{x_1}+\sqrt{x_2}\right)^2=x_1+x_2+2\sqrt{x_1x_2}=1+2\cdot\dfrac{1}{3}=\dfrac{5}{3}\)
=>\(\sqrt{x_1}+\sqrt{x_2}=\sqrt{\dfrac{5}{3}}\)
=>\(A=\dfrac{1}{3}\cdot\sqrt{\dfrac{5}{3}}=\dfrac{\sqrt{5}}{3\sqrt{3}}=\dfrac{\sqrt{15}}{9}\)
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