\(解方程:(m+n)^2 = (m-8)(n+8)\)
\[
\begin{align*}
m^2 +n^2 +2mn = mn +8m -8n -64 \\
\text{移项:} \\
m^2 +n^2 +2mm-mm-8m+8n +64 = 0 \\
\end{align*}
\]
\[
\text{这里才是关键,凑思维.} \\
\text{一:错误思维} \\
\]
\[
\begin{align*}
(m-4)^2 + (n+4)^2 +mm +32 =0 \\
\text{无法持续解.}
\end{align*}
\]
\[
\text{二:正确思维,两边同乘以2} \\
\]
\[
\begin{align*}
2 \cdot (m^2 +n^2 +2mm-mm-8m+8n +64) = 2 \cdot (0) \\
2m^2 +2n^2 +2mn -16m +16n+128 = 0 \\
(m-8)^2 -64 +(n+8)^2 – 64 +m^2 +n^2 +2mn +128 = 0 \\
(m-8)^2 -64 +(n+8)^2 – 64 +(m+n)^2 +128 =0 \\
(m-8)^2 +(n+8)^2 +(m+n)^2 =0 \\
\text{任何数的平方均大于等0,即方程有解需满足:} \\
\begin{cases}
(m-8)^2 = 0 \\
(n+8)^2 = 0 \\
(m+n)^2 = 0 \\
\end{cases}
\end{align*}
\]
\[
解得:m =8,n =-8.
\]