The difference of two numbers is 7. Their product is 18. What are the two numbers?

Sure, let's solve this step by step.

Step 1: Let's denote the two numbers as x and y. According to the problem, we know that:

x - y = 7 (equation 1)

and

x * y = 18 (equation 2)

Step 2: We can express y in terms of x using equation 1:

y = x - 7

Step 3: Substitute y in equation 2:

x * (x - 7) = 18

This simplifies to:

x^2 - 7x - 18 = 0

Step 4: This is a quadratic equation in the form 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 = -7, and c = -18.

Step 5: Calculate the discriminant (b^2 - 4ac):

(-7)^2 - 41(-18) = 49 + 72 = 121

Step 6: Since the discriminant is positive, we have two real solutions for x:

x1 = [7 + sqrt(121)] / (21) = 9 x2 = [7 - sqrt(121)] / (21) = -2

Step 7: Substitute x1 and x2 in equation 1 to find the corresponding y values:

For x1 = 9, y1 = 9 - 7 = 2 For x2 = -2, y2 = -2 - 7 = -9

So, the two pairs of numbers that satisfy the conditions are (9, 2) and (-2, -9).