Divide 12 into two parts such that the sum of their squares is 74.

Let one part be x.
Then the other part is 12 - x .
Condition:
Sum of the squares of two parts is 74.
Therefore x ² + (12 - x) ² = 74
x ² + (12 - x) ² = 74
x ² + 144 - 24x + x ² = 74
2x ² - 14x – 10x + 70 = 0
2x(x – 7) – 10(x – 7) = 0
(2x – 10)(x – 7) = 0
x = 5 or x = 7
Therefore the two products are 5 and 7.
Checking:
5 ² + 7 ² = 25 + 49 = 74.