The product of two numbers is 45 and their difference is 4 . The sum of squares of the two numbers is

Options

A. 135
B. 240
C. 73
D. 106

Answer

To solve the problem step by step, we will define the two numbers and use the information given to find the sum of their squares.

### Step 1: Define the Variables
Let the two numbers be \( X \) and \( Y \).

### Step 2: Set Up the Equations
From the problem statement, we have two equations:
1. The product of the two numbers:
\[ XY = 45 \]
2. The difference of the two numbers:
\[ X - Y = 4 \]

### Step 3: Express One Variable in Terms of the Other
From the second equation, we can express \( X \) in terms of \( Y \):
\[ X = Y + 4 \]

### Step 4: Substitute into the Product Equation
Now, substitute \( X \) in the product equation:
\[ (Y + 4)Y = 45 \]
Expanding this gives:
\[ Y^2 + 4Y = 45 \]

### Step 5: Rearrange the Equation
Rearranging the equation to set it to zero:
\[ Y^2 + 4Y - 45 = 0 \]

### Step 6: Factor the Quadratic Equation
Now we need to factor the quadratic equation:
\[ (Y + 9)(Y - 5) = 0 \]
This gives us two possible solutions for \( Y \):
\[ Y + 9 = 0 \quad \Rightarrow \quad Y = -9 \quad \text{(not valid since we are looking for positive numbers)} \]
\[ Y - 5 = 0 \quad \Rightarrow \quad Y = 5 \]

### Step 7: Find the Value of X
Using \( Y = 5 \) in the equation \( X = Y + 4 \):
\[ X = 5 + 4 = 9 \]

### Step 8: Calculate the Sum of Squares
Now we can find the sum of squares of \( X \) and \( Y \):
\[ X^2 + Y^2 = 9^2 + 5^2 = 81 + 25 = 106 \]

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