The sum of squares of two consecutive even natural numbers is 244; fin

1.	The sum of squares of two consecutive even natural numbers is 244; find the numbers.
Answer» Let the first even natural number be x. ∴ the next consecutive even natural number will be (x + 2). According to the given condition, x² + (x + 2)² = 244 ∴ x² + x² + 4x + 4 = 244 ∴ 2x² + 4x + 4 – 244 = 0 ∴ 2x² + 4x – 240 = 0 ∴ x² + 2x – 120 = 0 …[Dividing both sides by 2] ∴ x² + 12x – 10x – 120 = 0 ∴ x(x + 12) – 10 (x + 12) = 0 ∴ (x + 12) (x – 10) = 0 By using the property, if the product of two numbers is zero, then at least one of them is zero, we get ∴ x + 12 = 0 or x – 10 = 0 ∴ x = -12 or x = 10 But, natural number cannot be negative. ∴ x = 10 and x + 2 = 10 + 2 = 12 ∴ The two consecutive even natural numbers are 10 and 12.

The sum of squares of two consecutive even natural numbers is 244; find the numbers.

Answer»

Let the first even natural number be x.

∴ the next consecutive even natural number will be (x + 2).

According to the given condition,

x² + (x + 2)² = 244

∴ x² + x² + 4x + 4 = 244

∴ 2x² + 4x + 4 – 244 = 0

∴ 2x² + 4x – 240 = 0

∴ x² + 2x – 120 = 0 …[Dividing both sides by 2]

∴ x² + 12x – 10x – 120 = 0

∴ x(x + 12) – 10 (x + 12) = 0

∴ (x + 12) (x – 10) = 0

By using the property, if the product of two numbers is zero, then at least one of them is zero, we get

∴ x + 12 = 0 or x – 10 = 0

∴ x = -12 or x = 10

But, natural number cannot be negative.

∴ x = 10 and x + 2 = 10 + 2 = 12

∴ The two consecutive even natural numbers are 10 and 12.