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 step by step, we can follow these steps:

### Step 1: Define the Variables
Let the two positive numbers be \( a \) and \( b \).

### Step 2: Set Up the Equations
From the problem statement, we have two equations based on the conditions given:
1. The sum of the squares of the two numbers:
\[ a^2 + b^2 = 100 \quad \text{(Equation 1)} \]
2. The difference of the squares of the two numbers:
\[ a^2 - b^2 = 28 \quad \text{(Equation 2)} \]

### Step 3: Add the Two Equations
Now, we will add Equation 1 and Equation 2:
\[ (a^2 + b^2) + (a^2 - b^2) = 100 + 28 \]
This simplifies to:
\[ 2a^2 = 128 \]

### Step 4: Solve for \( a^2 \)
Now, divide both sides by 2:
\[ a^2 = 64 \]

### Step 5: Find \( a \)
Taking the square root of both sides, we find:
\[ a = \sqrt{64} = 8 \]

### Step 6: Substitute \( a \) Back to Find \( b^2 \)
Now, substitute \( a^2 = 64 \) back into Equation 1 to find \( b^2 \):
\[ 64 + b^2 = 100 \]
Subtracting 64 from both sides gives:
\[ b^2 = 36 \]

### Step 7: Find \( b \)
Taking the square root of both sides, we find:
\[ b = \sqrt{36} = 6 \]

### Step 8: Find the Sum of the Numbers
Now we can find the sum of the two numbers:
\[ a + b = 8 + 6 = 14 \]

### Final Answer
The sum of the two numbers is \( 14 \).

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