   SEARCH HOME Math Central Quandaries & Queries  Question from Lisa: a. Two positive numbers differ by 5. Their product is 234. What are the two numbers? b. The sum of two numbers is 12. Their product is 30. What are the two numbers? c.Two numbers differ by 5. Their squares differ by 55. What are the two numbers? d. The sum of two numbers is 15. Their product is 56. By forming an equation, find the two numbers. Hi Lisa,

Here is a similar problem that will show you how to approach your problems.

Two positive numbers differ by 4. Their product is 165. What are the two numbers?

There are two unknown numbers so let one be x and the other y. The two facts then become

Two positive numbers differ by 4: x - y = 4
Their product is 165: xy = 165

You want one equation in one unknown and you have two equations in two unknowns.

Since x - y = 4 adding y to both sides gives x = y + 4. Substitute this into the second equation to get

(y + 4)y = 165

which simplifies to

y2 + 4y - 165 = 0

The challenge is then to factor this quadratic. A little experimentation with the divisors of 165 gives me

y2 + 4y - 165 = (y + 15)(y - 11)

Thus y2 + 4y - 165 = 0 has solutions y = 11 and y = -15. Since we want y to be positive we have y = 11. Now from the first equation, x - y = 4, if y = 11 then x must be 15. Hence the two numbers are 11 and 15.     