The product of two numbers is 132, and their sum is 23. What are the two numbers?

This problem can be solved by using the method of solving quadratic equations.

Step 1: Let's denote the two numbers as x and y. From the problem, we have two equations:

x*y = 132 (equation 1) x + y = 23 (equation 2)

Step 2: From equation 2, we can express y as y = 23 - x.

Step 3: Substitute y in equation 1 with 23 - x, we get:

x * (23 - x) = 132 23x - x^2 = 132 x^2 - 23x + 132 = 0

Step 4: Now we have a quadratic equation in the form of ax^2 + bx + c = 0. We can solve it by using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a). In this case, a = 1, b = -23, c = 132.

Step 5: Calculate the discriminant (b^2 - 4ac) = (-23)^2 - 41132 = 529 - 528 = 1.

Step 6: Substitute a, b, and the discriminant into the quadratic formula, we get:

x = [23 ± sqrt(1)] / 2 So the two solutions are x = 12 and x = 11.

Therefore, the two numbers are 12 and 11.