Question

Factorise the following by taking out the common factors:
(mx + ny)² + (nx - my)²

Answer 1

(mx + ny)² + (nx - my)² 
= m²x² + n²y² + 2mnxy + n²x² + m²y² - 2mnxy
= m²x² + n²y² + n²x² + m²y² 
= m²x² + n²x² + m²y² + n²y²
= x²(m² + n²) + y²(m² + n²)
Here, the common factor is (m² + n²).
Dividing throughout by (m² + n²), we get

`(x^2("m"^2 + "n"^2))/(("m"^2 + "n"^2)) + (y^2("m"^2 + "n"^2))/(("m"^2 + "n"^2)`
= x² + y²
∴ (mx + ny)² + (nx - my)²
= (m² + n²)(x² + y²).