The sum of two numbers is 18 and their product is 45. Find the numbers using the method of factorization?

Let the two numbers be x and y. 

Now, the sum of the two numbers is 18. 

⇒ x + y = 18 

⇒ y = 18 − x... (1) 

Also, the product of the numbers is 45. 

⇒ xy = 45 

⇒ x(18 − x) = 45 .... [From (1)] 

⇒ 18x − x² = 45 

⇒ 18x − x² − 45 = 0 

⇒ − x² + 18x − 45 = 0 

⇒ x² − 18x + 45 = 0 

⇒ x² − (15 + 3)x + 45 = 0...... [(15 ×3) = 45 and (15 + 3) = 18] 

⇒ x² − 15x − 3x + 45 = 0 

⇒ x(x− 15) − 3(x − 15) = 0 

⇒ (x− 15)(x− 3) = 0 

∴ x− 15 = 0 or x− 3 = 0 

⇒ x = 15 or x = 3 

If x = 15, then y = 18 − 15 = 3 

If x = 3, then y = 18 − 3 = 15 

Thus, the two numbers are 3 and 15.