The number x is 2 more than the number y. If the sum of the squares of x and y is 34, find the product of x and y.

Given,

The number x is 2 more than the number y.

∴ x - y = 2

⇒ x = y + 2 ………..(1)

Given,

Sum of the squares of x and y is 34.

∴ x² + y² = 34

Substituting value of x from equation (1) in above equation, we get :

⇒ (y + 2)² + y² = 34

⇒ y² + 2² + 2 × y × 2 + y² = 34

⇒ 2y² + 4 + 4y = 34

⇒ 2y² + 4y + 4 - 34 = 0

⇒ 2y² + 4y - 30 = 0

⇒ 2(y² + 2y - 15) = 0

⇒ y² + 2y - 15 = 0

⇒ y² + 5y - 3y - 15 = 0

⇒ y(y + 5) - 3(y + 5) = 0

⇒ (y - 3)(y + 5) = 0

⇒ y - 3 = 0 or y + 5 = 0

⇒ y = 3 or y = -5.

If y = 3, x = y + 2 = 3 + 2 = 5, xy = 15.

If y = -5, x = y + 2 = -5 + 2 = -3, xy = 15.

Hence, xy = 15.