The sum of two numbers is 18 and their product is 56. Find the numbers.

Answer

To solve the problem of finding two numbers whose sum is 18 and product is 56, we can follow these steps:

### Step 1: Set up the equations
Let the two numbers be \( x \) and \( y \). According to the problem, we have:
1. \( x + y = 18 \) (Equation 1)
2. \( xy = 56 \) (Equation 2)

### Step 2: Express one variable in terms of the other
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 18 - x \]

### Step 3: Substitute into the product equation
Now, substitute \( y \) in Equation 2:
\[ x(18 - x) = 56 \]
This simplifies to:
\[ 18x - x^2 = 56 \]

### Step 4: Rearrange the equation
Rearranging the equation gives us:
\[ -x^2 + 18x - 56 = 0 \]
To make it a standard quadratic equation, we can multiply the entire equation by -1:
\[ x^2 - 18x + 56 = 0 \]

### Step 5: Factor the quadratic equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to 56 and add up to -18. The factors of 56 that satisfy this condition are -14 and -4:
\[ (x - 14)(x - 4) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x - 14 = 0 \) → \( x = 14 \)
2. \( x - 4 = 0 \) → \( x = 4 \)

### Step 7: Find corresponding \( y \) values
Now, we can find the corresponding \( y \) values using \( y = 18 - x \):
- If \( x = 14 \), then \( y = 18 - 14 = 4 \).
- If \( x = 4 \), then \( y = 18 - 4 = 14 \).

### Conclusion
The two numbers are \( 14 \) and \( 4 \).