The sum of the squares of two positive numbers is 100 and difference of their squares is 28. What is the sum of the given numbers.

Options

A. 12
B. 13
C. 14
D. 15

Answer

To solve the problem, we need to find two positive numbers \( x \) and \( y \) given the following conditions:

1. The sum of the squares of the two numbers is 100:
\[ x^2 + y^2 = 100 \]
2. The difference of their squares is 28:
\[ x^2 - y^2 = 28 \]

### Step 1: Set up the equations
We have the two equations:
1. \( x^2 + y^2 = 100 \) (Equation 1)
2. \( x^2 - y^2 = 28 \) (Equation 2)

### Step 2: Add the two equations
To eliminate \( y^2 \), we can add Equation 1 and Equation 2:
\[ (x^2 + y^2) + (x^2 - y^2) = 100 + 28 \]
This simplifies to:
\[ 2x^2 = 128 \]

### Step 3: Solve for \( x^2 \)
Now, divide both sides by 2:
\[ x^2 = 64 \]

### Step 4: Solve for \( x \)
Taking the square root of both sides gives:
\[ x = 8 \]

### Step 5: Substitute \( x \) back to find \( y \)
Now, substitute \( x \) back into Equation 1 to find \( y \):
\[ 8^2 + y^2 = 100 \]
This simplifies to:
\[ 64 + y^2 = 100 \]
Subtracting 64 from both sides gives:
\[ y^2 = 36 \]

### Step 6: Solve for \( y \)
Taking the square root of both sides gives:
\[ y = 6 \]

### Step 7: Find the sum of \( x \) and \( y \)
Now, we can find the sum of the two numbers:
\[ x + y = 8 + 6 = 14 \]

Thus, the sum of the given numbers is **14**.