\(已知a^2-b^2=9,ab=20,求2a+3b的值。\)
\[
\begin{align*}
& a^2-b^2=9 \\
& (a+b)(a-b)=9 \\
& [(a+b)(a-b)]^2 =9^2 \\
& (a+b)^2(a-b)^2 =81 \\
& (a^2+b^2+2ab)(a^2+b^2-2ab) =81 \\
& (a^2+b^2)^2-(2ab)^2 = 81 \\
& 已知ab=20 \\
& (a^2+b^2)^2-1600=81 \\
& (a^2+b^2)^2 = 1681 \\
& a^2+b^2 = 41 \\
& 已知a^2-b^2 = 9 \\
& 2a^2 = 50 \\
& a^2 = 25 \\
& a = \pm \sqrt{25} \\
& a = \pm 5 \\
& 求得 b = \mp 4 \\
& 2a+3b = \pm 22
\end{align*}
\]