If the sum of two numbers is 12 and their product is 32, what are the numbers?

If the sum of two numbers is 12 and their product is 32, what are the numbers?

Step 1: Let the two numbers be x and y.
Step 2: Write down the first equation based on the sum: x + y = 12.
Step 3: Write down the second equation based on the product: x * y = 32.
Step 4: From the first equation, express y in terms of x: y = 12 - x.
Step 5: Substitute y in the second equation: x * (12 - x) = 32.
Step 6: Expand the equation: 12x - x^2 = 32.
Step 7: Rearrange the equation to form a standard quadratic equation: x^2 - 12x + 32 = 0.
Step 8: Factor the quadratic equation: (x - 4)(x - 8) = 0.
Step 9: Solve for x: x = 4 or x = 8.
Step 10: Use the values of x to find y: If x = 4, then y = 12 - 4 = 8; If x = 8, then y = 12 - 8 = 4.
Step 11: The two numbers are 4 and 8.

Algebraic Equations – The question tests the ability to set up and solve a system of equations based on given conditions (sum and product of two numbers).
Quadratic Equations – The problem can be transformed into a quadratic equation, testing knowledge of factoring or using the quadratic formula.