Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two by 60.

Assume the middle number to be x and form a quadratic equation satisfying the above statement. Hence; find the three numbers.

Let three consecutive numbers be (x - 1), x and (x + 1).

According to question,

⇒ x² - [(x + 1)² - (x - 1)²] = 60

⇒ x² - [x² + 1 + 2x - (x² + 1 - 2x)] = 60

⇒ x² - [x² + 1 + 2x - x² - 1 + 2x] = 60

⇒ x² - 4x = 60

⇒ x² - 4x - 60 = 0

⇒ x² - 10x + 6x - 60 = 0

⇒ x(x - 10) + 6(x + 10) = 0

⇒ (x + 6)(x - 10) = 0

⇒ x + 6 = 0 or x - 10 = 0

⇒ x = -6 or x = 10.

Since, numbers are natural numbers,

∴ x = 10.

∴ x - 1 = 9 and x + 1 = 11.

Hence, the numbers are 9, 10, 11 and quadratic equation = x² - 4x - 60 = 0.