Two numbers differ by 8, and their product is 240. Find

Solution

Let the numbers be x and y, with x > y. We’re told: x − y = 8 x·y = 240 From (1), x = y + 8. Plug that into (2): (y + 8)·y = 240 y² + 8y = 240 y² + 8y − 240 = 0 Now solve the quadratic: y² + 8y − 240 = 0 We look for two numbers that multiply to −240 and add to +8. Those numbers are +20 and −12, because: 20 × (−12) = −240 20 + (−12) = 8 So we can factor: (y + 20)(y − 12) = 0 That gives two possible y values: y = −20 or y = 12 Now find x = y + 8 in each case: If y = 12 x = 12 + 8 = 20 Pair: (20, 12) If y = −20 x = −20 + 8 = −12 Pair: (−12, −20) Check both: 20 − 12 = 8 and 20 × 12 = 240 ✓ (−12) − (−20) = 8 and (−12) × (−20) = 240 ✓ Ignore the negatuive value we get, Final answer: The numbers are 20 and 12