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