Two numbers have a sum of 13. The sum of their squares is 85. What is the product of these two numbers?

Let's solve this problem step by step.

Step 1: Let's assume the two numbers as x and y.

Step 2: According to the given information, the sum of the two numbers is 13. So, we can write the equation as: x + y = 13

Step 3: The sum of their squares is given as 85. So, we can write the equation as: x^2 + y^2 = 85

Step 4: Now, we have a system of two equations with two variables. We can solve this system of equations to find the values of x and y.

Step 5: We can use the method of substitution or elimination to solve the system of equations. Let's use the method of substitution.

Step 6: From the first equation, we can express y in terms of x as: y = 13 - x

Step 7: Substitute this value of y in the second equation: x^2 + (13 - x)^2 = 85

Step 8: Simplify the equation: x^2 + 169 - 26x + x^2 = 85

Step 9: Combine like terms: 2x^2 - 26x + 84 = 0

Step 10: Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use factoring.

Step 11: Factor the quadratic equation: 2(x^2 - 13x + 42) = 0

Step 12: Set each factor equal to zero and solve for x: x^2 - 13x + 42 = 0

(x - 6)(x - 7) = 0

x - 6 = 0 or x - 7 = 0

x = 6 or x = 7

Step 13: Now that we have the values of x, we can substitute them back into the first equation to find the corresponding values of y.

For x = 6: y = 13 - x y = 13 - 6 y = 7

For x = 7: y = 13 - x y = 13 - 7 y = 6

Step 14: The two numbers are 6 and 7.

Step 15: Finally, to find the product of these two numbers, we multiply them: Product = 6 * 7 Product = 42

Therefore, the product of the two numbers is 42.