The difference between two positive numbers is 5 and the sum of their squares is 73. Find the product of these numbers.

Let two positive numbers be x and y.

Given,

Difference between two positive numbers is 5.

∴ x - y = 5

⇒ x = y + 5 ………..(1)

Given,

Sum of squares of numbers is 73.

∴ x² + y² = 73

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

⇒ (y + 5)² + y² = 73

⇒ y² + 5² + 2 × y × 5 + y² = 73

⇒ 2y² + 25 + 10y = 73

⇒ 2y² + 10y + 25 - 73 = 0

⇒ 2y² + 10y - 48 = 0

⇒ 2(y² + 5y - 24) = 0

⇒ y² + 5y - 24 = 0

⇒ y² + 8y - 3y - 24 = 0

⇒ y(y + 8) - 3(y + 8) = 0

⇒ (y - 3)(y + 8) = 0

⇒ y - 3 = 0 or y + 8 = 0

⇒ y = 3 or y = -8.

If y = 3, x = y + 5 = 3 + 5 = 8, xy = 24,

If y = -8, x = y + 5 = -8 + 5 = -3, xy = 24.

Hence, product of numbers = 24.