The difference between two positive numbers is 3. If the sum of their squares is 369, then the sum of the numbers is :

Options

A. 81
B. 33
C. 27
D. 25

Answer

To solve the problem step by step, we start by defining the two positive numbers. Let's denote them as \( x \) and \( y \).

### Step 1: Set up the equations
From the problem, we know:
1. The difference between the two numbers is 3:
\[ x - y = 3 \quad \text{(1)} \]
2. The sum of their squares is 369:
\[ x^2 + y^2 = 369 \quad \text{(2)} \]

### Step 2: Express one variable in terms of the other
From equation (1), we can express \( x \) in terms of \( y \):
\[ x = y + 3 \quad \text{(3)} \]

### Step 3: Substitute into the second equation
Now, we substitute equation (3) into equation (2):
\[ (y + 3)^2 + y^2 = 369 \]

### Step 4: Expand and simplify
Expanding \( (y + 3)^2 \):
\[ y^2 + 6y + 9 + y^2 = 369 \]
Combining like terms:
\[ 2y^2 + 6y + 9 = 369 \]

### Step 5: Rearrange the equation
Now, we rearrange the equation to set it to zero:
\[ 2y^2 + 6y + 9 - 369 = 0 \]
\[ 2y^2 + 6y - 360 = 0 \]

### Step 6: Simplify the quadratic equation
We can divide the entire equation by 2 to simplify:
\[ y^2 + 3y - 180 = 0 \]

### Step 7: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to \(-180\) and add to \(3\). The numbers \(15\) and \(-12\) satisfy this:
\[ (y + 15)(y - 12) = 0 \]

### Step 8: Solve for \( y \)
Setting each factor to zero gives us:
\[ y + 15 = 0 \quad \Rightarrow \quad y = -15 \quad \text{(not valid since y is positive)} \]
\[ y - 12 = 0 \quad \Rightarrow \quad y = 12 \]

### Step 9: Find \( x \)
Now, we substitute \( y = 12 \) back into equation (3) to find \( x \):
\[ x = y + 3 = 12 + 3 = 15 \]

### Step 10: Find the sum of the numbers
Finally, we find the sum of the two numbers:
\[ x + y = 15 + 12 = 27 \]

### Conclusion
The sum of the two numbers is \( \boxed{27} \).

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