Divide 12 into two parts such that the sum of their squares is greater than twice their product by 4. The parts are?
Divide 12 into two parts such that the sum of their squares is greater than twice their product by 4. The parts are?
Explanation
Let's denote the two parts as x and 12 - x.
The sum of their squares is x² + (12 - x)².
Twice their product is 2x(12 - x).
According to the problem:
x² + (12 - x)² - 2x(12 - x) = 4
x² + 144 - 24x + x² - 24x + 2x² = 4
4x² - 48x + 140 = 0
x² - 12x + 35 = 0
(x - 7)(x - 5) = 0
x = 7 or x = 5
The two parts are 5 and 7.
Let's check:
5² + 7² = 25 + 49 = 74
2 × 5 × 7 = 70
74 - 70 = 4
This satisfies the condition.
The answer is 5 and 7.