The difference between two numbers is 14 and the difference between their squares is 448. Find the numbers.

Explanation

To solve the problem step by step, we will define the two numbers and set up equations based on the information given in the question.

### Step 1: Define the variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set up the equations
According to the problem, we have two pieces of information:
1. The difference between the two numbers is 14:
\[ x - y = 14 \quad \text{(Equation 1)} \]

2. The difference between their squares is 448:
\[ x^2 - y^2 = 448 \quad \text{(Equation 2)} \]

### Step 3: Use the difference of squares
We can use the identity \( a^2 - b^2 = (a - b)(a + b) \) on Equation 2:
\[ x^2 - y^2 = (x - y)(x + y) \]
Substituting Equation 1 into this gives:
\[ 448 = (14)(x + y) \]

### Step 4: Solve for \( x + y \)
Now, we can solve for \( x + y \):
\[ x + y = \frac{448}{14} = 32 \quad \text{(Equation 3)} \]

### Step 5: Solve the system of equations
Now we have a system of two equations:
1. \( x - y = 14 \) (Equation 1)
2. \( x + y = 32 \) (Equation 3)

We can add these two equations:
\[ (x - y) + (x + y) = 14 + 32 \]
This simplifies to:
\[ 2x = 46 \]
So,
\[ x = \frac{46}{2} = 23 \]

### Step 6: Find \( y \)
Now, substitute \( x = 23 \) back into Equation 1 to find \( y \):
\[ 23 - y = 14 \]
This simplifies to:
\[ y = 23 - 14 = 9 \]

### Conclusion
The two numbers are:
\[ x = 23 \quad \text{and} \quad y = 9 \]