The sum of two positive numbers is 14 and difference between their squares is 56. What is the sum of their squares ?

Options

A. 106
B. 196
C. 53
D. 68

Answer

To solve the problem step by step, let's denote the two positive numbers as \( a \) and \( b \).

### Step 1: Set up the equations based on the problem statement.
According to the problem, we have two pieces of information:
1. The sum of the two numbers:
\[ a + b = 14 \quad \text{(Equation 1)} \]
2. The difference between their squares:
\[ a^2 - b^2 = 56 \quad \text{(Equation 2)} \]

### Step 2: Use the difference of squares formula.
The difference of squares can be factored using the identity \( a^2 - b^2 = (a + b)(a - b) \). Therefore, we can rewrite Equation 2 as:
\[ (a + b)(a - b) = 56 \]
Substituting Equation 1 into this gives:
\[ 14(a - b) = 56 \]

### Step 3: Solve for \( a - b \).
Now, we can solve for \( a - b \):
\[ a - b = \frac{56}{14} = 4 \quad \text{(Equation 3)} \]

### Step 4: Solve the system of equations.
Now we have two equations:
1. \( a + b = 14 \) (Equation 1)
2. \( a - b = 4 \) (Equation 3)

We can add these two equations to eliminate \( b \):
\[ (a + b) + (a - b) = 14 + 4 \]
This simplifies to:
\[ 2a = 18 \]
Thus, we find:
\[ a = \frac{18}{2} = 9 \]

### Step 5: Find the value of \( b \).
Now, substitute \( a = 9 \) back into Equation 1 to find \( b \):
\[ 9 + b = 14 \]
This gives:
\[ b = 14 - 9 = 5 \]

### Step 6: Calculate the sum of their squares.
Now that we have both numbers, \( a = 9 \) and \( b = 5 \), we can find the sum of their squares:
\[ a^2 + b^2 = 9^2 + 5^2 = 81 + 25 = 106 \]

### Final Answer:
The sum of their squares is:
\[ \boxed{106} \]