The sum of two numbers is 437 and their product is 21982. Find the numbers.
A. 399 and 38
B. 295 and 142
C. 58 and 379
D. 323 and 114

Options

A. A
B. B
C. C
D. D

Answer

To solve the problem, we need to find two numbers whose sum is 437 and whose product is 21,982. Let's denote the two numbers as \( x \) and \( y \).

### Step 1: Set up the equations
We have the following two equations based on the problem statement:
1. \( x + y = 437 \) (Equation 1)
2. \( xy = 21,982 \) (Equation 2)

### Step 2: Use the identity for squares
We can use the identity:
\[ (x - y)^2 = (x + y)^2 - 4xy \]
Substituting the values from our equations:
\[ (x - y)^2 = (437)^2 - 4 \times 21,982 \]

### Step 3: Calculate \( (x + y)^2 \)
First, calculate \( (437)^2 \):
\[ 437^2 = 190,969 \]

### Step 4: Calculate \( 4xy \)
Now calculate \( 4 \times 21,982 \):
\[ 4 \times 21,982 = 87,928 \]

### Step 5: Substitute values into the identity
Now substitute these values into the identity:
\[ (x - y)^2 = 190,969 - 87,928 \]
Calculating the right-hand side:
\[ (x - y)^2 = 103,041 \]

### Step 6: Take the square root
Now, take the square root of both sides to find \( x - y \):
\[ x - y = \sqrt{103,041} = 321 \]

### Step 7: Set up a new equation
Now we have two equations:
1. \( x + y = 437 \) (Equation 1)
2. \( x - y = 321 \) (Equation 3)

### Step 8: Add the equations
Add Equation 1 and Equation 3:
\[ (x + y) + (x - y) = 437 + 321 \]
This simplifies to:
\[ 2x = 758 \]
So, divide by 2:
\[ x = 379 \]

### Step 9: Find \( y \)
Now substitute \( x \) back into Equation 1 to find \( y \):
\[ 379 + y = 437 \]
Subtract 379 from both sides:
\[ y = 437 - 379 = 58 \]

### Conclusion
The two numbers are \( x = 379 \) and \( y = 58 \).

### Final Answer
The numbers are **58 and 379** (Option C).

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