Question

The sum of the squares of two consecutive even natural numbers is 100. Find the numbers.

Answer 1

Let the two consecutive even natural numbers be x and (x + 2). Then, by the given condition, we have
x² + (x + 2)² = 100
$\Rightarrow$ x² + x² + 4x + 4 = 100
$\Rightarrow$ 2x² + 4x – 96 = 0
$\Rightarrow$ 2(x² + 2x – 48) = 0
$\Rightarrow$ x² + 2x – 48 = 0
On splitting the middle term 2x as 8x – 6x, we get:
x² + 8x – 6x – 48 = 0
$\Rightarrow$ x(x + 8) – 6(x + 8) = 0
$\Rightarrow$ (x + 8)(x – 6) = 0
$\Rightarrow$ x + 8 = 0 or x – 6 = 0
$\Rightarrow$ x = –8 or x = 6
Since x is an even natural number, it cannot be negative.
Therefore, x = 6 and (x + 2) = 8.
Thus, the two consecutive even natural numbers are 6 and 8.