The product of two numbers is 144, and their sum is 24. 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 know that:

x*y = 144 (equation 1) x + y = 24 (equation 2)

Step 2: From equation 2, we can express y as y = 24 - x.

Step 3: Substitute y in equation 1 with 24 - x, we get:

x * (24 - x) = 144 24x - x^2 = 144 x^2 - 24x + 144 = 0

Step 4: This is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 1, b = -24, and c = 144. We can solve it using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a).

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

x = [24 ± sqrt((-24)^2 - 41144)] / (2*1) x = [24 ± sqrt(576 - 576)] / 2 x = [24 ± 0] / 2

Step 6: So the solutions are x = 24/2 = 12 and x = 24/2 = 12.

Step 7: Substitute x = 12 into equation 2, we get y = 24 - 12 = 12.

So the two numbers are 12 and 12.