The product of two numbers is 54, and their sum is 15. What are the two numbers?

To solve this problem, we can use 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 = 54 (equation 1) x + y = 15 (equation 2)

Step 2: From equation 2, we can express y as y = 15 - x.

Step 3: Substitute y in equation 1 with 15 - x, we get:

x * (15 - x) = 54 15x - x^2 = 54 x^2 - 15x + 54 = 0

Step 4: Now we have a quadratic equation in the form of ax^2 + bx + c = 0. We can solve it using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a). In this case, a = 1, b = -15, c = 54.

Step 5: Substitute a, b, c into the quadratic formula, we get:

x = [15 ± sqrt((-15)^2 - 4154)] / (2*1) x = [15 ± sqrt(225 - 216)] / 2 x = [15 ± sqrt(9)] / 2 x = [15 ± 3] / 2

Step 6: So the solutions are x = 9 and x = 6. Substitute x into equation 2, we can get the corresponding y values, which are also 9 and 6.

So the two numbers are 9 and 6.