Simplify the expression:

$$(5z - 7y)^2 + (7z + 5y)^2 - 49z^2$$

Question

Simplify the expression:

$$(5z - 7y)^2 + (7z + 5y)^2 - 49z^2$$

Accepted Answer

**Solution:**

1. Expand each square using the identity:
   - $(a - b)^2 = a^2 - 2ab + b^2$
   - $(a + b)^2 = a^2 + 2ab + b^2$

2. Expand $(5z - 7y)^2$:
   $$25z^2 - 2 	imes 5z 	imes 7y + 49y^2 = 25z^2 - 70zy + 49y^2$$

3. Expand $(7z + 5y)^2$:
   $$49z^2 + 2 	imes 7z 	imes 5y + 25y^2 = 49z^2 + 70zy + 25y^2$$

4. Substitute these into the original expression:
   $$25z^2 - 70zy + 49y^2 + 49z^2 + 70zy + 25y^2 - 49z^2$$

5. Combine like terms:
   - The $-70zy$ and $+70zy$ cancel out.
   - Combine $z^2$ terms: $25z^2 + 49z^2 - 49z^2 = 25z^2$
   - Combine $y^2$ terms: $49y^2 + 25y^2 = 74y^2$

6. Final simplified expression:
   $$25z^2 + 74y^2$$

**Answer:** $25z^2 + 74y^2$

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