If `a^(2) + b^(2) + 2b + 4a + 5 = 0` , then the value of `(a - b)/( a + b)` is

Options

A. 3
B. `-3`
C. `(1)/(3)`
D. `-(1)/(3)`

Answer

To solve the equation \( a^2 + b^2 + 2b + 4a + 5 = 0 \) and find the value of \( \frac{a - b}{a + b} \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ a^2 + b^2 + 2b + 4a + 5 = 0 \]
We can rearrange the terms to group \( a \) and \( b \):
\[ a^2 + 4a + b^2 + 2b + 5 = 0 \]

### Step 2: Completing the Square for \( a \)
Next, we complete the square for the \( a \) terms:
\[ a^2 + 4a = (a + 2)^2 - 4 \]
Thus, substituting back into the equation gives:
\[ (a + 2)^2 - 4 + b^2 + 2b + 5 = 0 \]

### Step 3: Completing the Square for \( b \)
Now, we complete the square for the \( b \) terms:
\[ b^2 + 2b = (b + 1)^2 - 1 \]
Substituting this into the equation results in:
\[ (a + 2)^2 - 4 + (b + 1)^2 - 1 + 5 = 0 \]
This simplifies to:
\[ (a + 2)^2 + (b + 1)^2 = 0 \]

### Step 4: Analyzing the Equation
Since the sum of two squares is zero, both squares must individually be zero:
\[ (a + 2)^2 = 0 \quad \text{and} \quad (b + 1)^2 = 0 \]
This gives us:
\[ a + 2 = 0 \implies a = -2 \]
\[ b + 1 = 0 \implies b = -1 \]

### Step 5: Finding \( \frac{a - b}{a + b} \)
Now we can substitute \( a \) and \( b \) into the expression \( \frac{a - b}{a + b} \):
\[ \frac{a - b}{a + b} = \frac{-2 - (-1)}{-2 + (-1)} = \frac{-2 + 1}{-2 - 1} = \frac{-1}{-3} \]
This simplifies to:
\[ \frac{1}{3} \]

### Final Answer
Thus, the value of \( \frac{a - b}{a + b} \) is:
\[ \frac{1}{3} \]

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