Two positive numbers difference by 5. Three times the square of the larger number exceeds twice the square of smaller number by 334, find the larger of these two numbers.

It is given that difference between two positive numbers = 5.

Let the one number be x and other number be 5 + x.

Three times the square of the larger number exceeds twice the square of smaller number by 334.

⇒ 3 $\times$ (5 + x)² - 2 $\times$ x² = 334

⇒ 3 $\times$ (25 + x² + 10x) - 2 $\times$ x² = 334

⇒ 75 + 3x² + 30x - 2x² - 334 = 0

⇒ x² + 30x - 259 = 0

⇒ x² + 37x - 7x - 259 = 0

⇒ x(x + 37) - 7(x + 37) = 0

⇒ (x + 37)(x - 7) = 0

⇒ (x + 37) = 0 or (x - 7) = 0

⇒ x = -37 or x = 7

Larger number = 5 + x = 5 + 7 = 12

Hence, the larger number = 12.