Two positive numbers have a difference of 8 and a product of 33. What are these numbers?

To find the two positive numbers, we’ll set up a couple of equations based on the information given.

Let’s define the two numbers as x and y. According to the problem:

• The difference between the two numbers is 8: x – y = 8
• The product of the two numbers is 33: xy = 33

From the first equation, we can express x in terms of y:

x = y + 8

Now, we can substitute this expression for x into the second equation:

(y + 8)y = 33

This simplifies to:

y² + 8y – 33 = 0

Next, we will use the quadratic formula to solve for y:

y = (-b ± √(b² – 4ac)) / 2a

Substituting a = 1, b = 8, and c = -33 into the formula, we have:

y = (-8 ± √(8² – 4 × 1 × -33)) / (2 × 1)

This results in:

y = (-8 ± √(64 + 132)) / 2

y = (-8 ± √196) / 2

y = (-8 ± 14) / 2

Calculating the two possible values for y:

• y = (6) / 2 = 3
• y = (-22) / 2 (not valid since we’re looking for positive numbers)

Thus, we find y = 3. Now, substituting back to find x:

x = y + 8 = 3 + 8 = 11

Therefore, the two positive numbers are 3 and 11, which satisfy both conditions of having a difference of 8 and a product of 33.