If the difference of two numbers is 3 and the difference of their squares is 39. then the larger number is

Options

A. 8
B. 9
C. 12
D. 13

Answer

To solve the problem, we need to find two numbers based on the given conditions:

1. The difference of the two numbers is 3.
2. The difference of their squares is 39.

Let's denote the two numbers as \( x \) (the larger number) and \( y \) (the smaller number).

### Step 1: Set up the equations
From the problem, we can set up the following equations based on the given information:

1. \( x - y = 3 \) (Equation 1)
2. \( x^2 - y^2 = 39 \) (Equation 2)

### Step 2: Use the difference of squares formula
We can use the difference of squares formula, which states that \( a^2 - b^2 = (a - b)(a + b) \). Applying this to our second equation:

\[ x^2 - y^2 = (x - y)(x + y) \]

Substituting Equation 1 into this gives us:

\[ 39 = (3)(x + y) \]

### Step 3: Solve for \( x + y \)
Now, we can solve for \( x + y \):

\[ 39 = 3(x + y) \]

Dividing both sides by 3:

\[ x + y = \frac{39}{3} = 13 \quad \text{(Equation 3)} \]

### Step 4: Solve the system of equations
Now we have two equations:

1. \( x - y = 3 \) (Equation 1)
2. \( x + y = 13 \) (Equation 3)

We can solve these equations simultaneously.

Adding Equation 1 and Equation 3:

\[ (x - y) + (x + y) = 3 + 13 \]
\[ 2x = 16 \]

Dividing both sides by 2 gives:

\[ x = 8 \]

### Step 5: Find \( y \)
Now, we can substitute \( x = 8 \) back into Equation 1 to find \( y \):

\[ 8 - y = 3 \]

Rearranging gives:

\[ y = 8 - 3 = 5 \]

### Conclusion
The larger number \( x \) is 8.

### Final Answer
The larger number is **8**.

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