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

2. The difference of their squares:
\[ x^2 - y^2 = 28 \]

### Step 1: Set up the equations

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

### Step 2: Add the two equations

Adding 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 \( x^2 \)

Dividing both sides by 2:
\[ x^2 = \frac{128}{2} = 64 \]

### Step 4: Solve for \( x \)

Taking the square root of both sides:
\[ x = \sqrt{64} = 8 \]

### Step 5: Substitute \( x \) back to find \( y^2 \)

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

### Step 6: Solve for \( y \)

Taking the square root of both sides:
\[ y = \sqrt{36} = 6 \]

### Step 7: Find the sum of the numbers

Now, we can find the sum of the two numbers:
\[ x + y = 8 + 6 = 14 \]

### Final Answer

The sum of the two numbers is \( \boxed{14} \).

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