The difference of two positive numbers is six. Their product is 223 less than the sum of their squares. What are the numbers?

Question

Algebra 1

Answer 1

Let the numbers be a nd b.

Given the following equations:

a ~ b = 6 –––––––––– (1)

a⋅b = a² + b² - 223 –––––––––– (2)

(a ~ b)²

= a² + b² - 2⋅a⋅b

= a⋅b + 223 - 2⋅a⋅b [from equation (2)]

= 223 - a⋅b

∴ (a ~ b)² = 223 - a⋅b

⇒ 6² = 223 - a⋅b [from equation (1)]

⇒ a⋅b = 187

187 has four factors: 1, 11, 17, 187

We can rule 1 and 187 out as 1 ~ 187 ≠ 6

∴ The numbers are 11 and 17.