If `A^(2) -B^(2) = 864 and A+B = 36` then find the value of 'B'?

Options

A. 4
B. 6
C. 7
D. 8

Explanation

To solve the equations \( A^2 - B^2 = 864 \) and \( A + B = 36 \), we will follow these steps:

### Step 1: Rewrite the first equation using the difference of squares
The equation \( A^2 - B^2 \) can be factored using the difference of squares formula:
\[ A^2 - B^2 = (A + B)(A - B) \]
Given that \( A + B = 36 \), we can substitute this into the equation:
\[ (A + B)(A - B) = 864 \]
This becomes:
\[ 36(A - B) = 864 \]

### Step 2: Solve for \( A - B \)
Now, we can solve for \( A - B \) by dividing both sides of the equation by 36:
\[ A - B = \frac{864}{36} \]
Calculating the right side:
\[ A - B = 24 \]

### Step 3: Set up a system of equations
Now we have two equations:
1. \( A + B = 36 \)
2. \( A - B = 24 \)

### Step 4: Solve the system of equations
We can add these two equations together to eliminate \( B \):
\[ (A + B) + (A - B) = 36 + 24 \]
This simplifies to:
\[ 2A = 60 \]
Now, divide by 2 to find \( A \):
\[ A = 30 \]

### Step 5: Substitute to find \( B \)
Now that we have \( A \), we can substitute it back into one of the original equations to find \( B \). We will use \( A + B = 36 \):
\[ 30 + B = 36 \]
Subtracting 30 from both sides gives us:
\[ B = 6 \]

### Final Answer
The value of \( B \) is \( 6 \).

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