The sum of the squares of three consecutive even numbers is 596. Find the numbers.

Let the three consecutive even numbers be represented as x, x + 2, and x + 4.

Given,

The sum of their squares is 596.

⇒ x² + (x + 2)² + (x + 4)² = 596

⇒ x² + x² + 4x + 4 + x² + 8x + 16 = 596

⇒ 3x² + 12x + 20 = 596

⇒ 3x² + 12x + 20 - 596 = 0

⇒ 3x² + 12x - 576 = 0

⇒ 3(x² + 4x − 192) = 0

⇒ x² + 4x − 192 = 0

⇒ x² + 16x - 12x − 192 = 0

⇒ x(x + 16) - 12(x + 16) = 0

⇒ (x - 12)(x + 16) = 0

⇒ (x - 12) = 0 or (x + 16) = 0 [Using zero product rule]

⇒ x = 12 or x = -16.

Case 1 : x = 12

The three consecutive even numbers are:

⇒ x = 12

⇒ x + 2 = 12 + 2 = 14

⇒ x + 4 = 12 + 4 = 16

Case 2: x = −16

The three consecutive even numbers are:

⇒ x = −16

⇒ x + 2 = −16 + 2 = −14

⇒ x + 4 = −16 + 4 = −12

Hence, the two possible sets of three consecutive even numbers are 12, 14, 16 and -16, -14, -12.