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(a-b)^2 Expansion and Visualisation

Simplify: 7m-...

Question

Simplify: ${(7 m - 8 n)}^{2} + {(7 m + 8 n)}^{2}$ .

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Solution

Step 1. Expand ${(7 m - 8 n)}^{2}$ .
Expand the term ${(7 m - 8 n)}^{2}$ :
$\begin{matrix} {(7 m - 8 n)}^{2} = {(7 m)}^{2} + {(8 n)}^{2} - 2 (7 m) (8 n) \\ = 49 m^{2} + 64 n^{2} - 112 m n \end{matrix}$
Step 2: Expand ${(7 m + 8 n)}^{2}$ .
Expand the term ${(7 m + 8 n)}^{2}$ :
$\begin{matrix} {(7 m + 8 n)}^{2} = {(7 m)}^{2} + {(8 n)}^{2} + 2 (7 m) (8 n) \\ = 49 m^{2} + 64 n^{2} + 112 m n \end{matrix}$
Step 3. Simplify ${(7 m - 8 n)}^{2} + {(7 m + 8 n)}^{2}$ .
Add the expanded terms of ${(7 m - 8 n)}^{2} + {(7 m + 8 n)}^{2}$ :
$\begin{matrix} {(7 m - 8 n)}^{2} + {(7 m + 8 n)}^{2} = [49 m^{2} + 64 n^{2} - 112 m n] + [49 m^{2} + 64 n^{2} + 112 m n] \\ = 98 m^{2} + 128 n^{2} \end{matrix}$
Hence, the value of the given expression is ${(7 m - 8 n)}^{2} + {(7 m + 8 n)}^{2} = 41 m^{2} + 41 n^{2} + 80 m n$

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