If sum of two numbers is 42 and their product is 437, then find their difference.

Options

A. 3
B. 4
C. 5
D. 7

Answer

To solve the problem step by step, we will use the information given about the sum and product of two numbers.

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

### Step 2: Write the equations based on the problem
From the problem, we know:
1. The sum of the two numbers:
\[ A + B = 42 \]
2. The product of the two numbers:
\[ AB = 437 \]

### Step 3: Use the identity for the square of a sum
We can use the identity:
\[ (A + B)^2 = A^2 + B^2 + 2AB \]
Substituting the known values:
\[ 42^2 = A^2 + B^2 + 2 \times 437 \]

### Step 4: Calculate \( 42^2 \) and \( 2 \times 437 \)
Calculating \( 42^2 \):
\[ 42^2 = 1764 \]
Calculating \( 2 \times 437 \):
\[ 2 \times 437 = 874 \]

### Step 5: Substitute these values into the equation
Now substituting back into the equation:
\[ 1764 = A^2 + B^2 + 874 \]

### Step 6: Solve for \( A^2 + B^2 \)
Rearranging the equation gives:
\[ A^2 + B^2 = 1764 - 874 \]
Calculating the right-hand side:
\[ A^2 + B^2 = 890 \]

### Step 7: Use the identity for the square of a difference
Now we can use the identity:
\[ (A - B)^2 = A^2 + B^2 - 2AB \]
Substituting the values we have:
\[ (A - B)^2 = 890 - 874 \]

### Step 8: Calculate \( (A - B)^2 \)
Calculating the right-hand side:
\[ (A - B)^2 = 16 \]

### Step 9: Take the square root to find \( A - B \)
Taking the square root gives:
\[ A - B = \sqrt{16} = 4 \]

### Conclusion
Thus, the difference between the two numbers \( A \) and \( B \) is:
\[ \boxed{4} \]