Suppose `a ^(2) + 2ab + b^(2) =c ^(2) and c -b =4.` Assuming `c gt 0,` which is the value of a ?

Explanation

To solve the problem step by step, we start with the given equations:

1. **Given Equations**:
\[ a^2 + 2ab + b^2 = c^2 \]
\[ c - b = 4 \]

2. **Recognizing the Identity**:
The left-hand side of the first equation can be recognized as a perfect square:
\[ a^2 + 2ab + b^2 = (a + b)^2 \]
Therefore, we can rewrite the equation as:
\[ (a + b)^2 = c^2 \]

3. **Taking the Square Root**:
Taking the square root of both sides, we have:
\[ a + b = c \quad \text{or} \quad a + b = -c \]
However, since \(c > 0\), we only consider:
\[ a + b = c \]

4. **Substituting for \(c\)**:
From the second equation, we know:
\[ c = b + 4 \]
Now we can substitute this value of \(c\) into the equation \(a + b = c\):
\[ a + b = b + 4 \]

5. **Solving for \(a\)**:
To isolate \(a\), we subtract \(b\) from both sides:
\[ a = 4 \]

Thus, the value of \(a\) is:
\[ \boxed{4} \]