If ` a^(2) + b^(2) = 4 b + 6a - 13` then what is the value of `(a + b)` ?

Options

A. 3
B. 2
C. 5
D. 10

Answer

To solve the equation \( a^2 + b^2 = 4b + 6a - 13 \) and find the value of \( a + b \), we can follow these steps:

### Step 1: Rearrange the equation
We start by moving all terms to one side of the equation:

\[ a^2 + b^2 - 4b - 6a + 13 = 0 \]

### Step 2: Group the terms
Next, we can group the terms related to \( a \) and \( b \):

\[ a^2 - 6a + b^2 - 4b + 13 = 0 \]

### Step 3: Complete the square for \( a \)
To complete the square for the \( a \) terms, we take the coefficient of \( a \) (which is -6), halve it to get -3, and square it to get 9. We rewrite the \( a \) terms:

\[ (a^2 - 6a + 9) + b^2 - 4b + 13 - 9 = 0 \]

This simplifies to:

\[ (a - 3)^2 + b^2 - 4b + 4 = 0 \]

### Step 4: Complete the square for \( b \)
Now we complete the square for the \( b \) terms. The coefficient of \( b \) is -4, halve it to get -2, and square it to get 4. We rewrite the \( b \) terms:

\[ (a - 3)^2 + (b^2 - 4b + 4) = 0 \]

This simplifies to:

\[ (a - 3)^2 + (b - 2)^2 = 0 \]

### Step 5: Analyze the equation
Since the sum of two squares equals zero, both squares must individually equal zero:

\[ (a - 3)^2 = 0 \quad \text{and} \quad (b - 2)^2 = 0 \]

### Step 6: Solve for \( a \) and \( b \)
From \( (a - 3)^2 = 0 \), we find:

\[ a - 3 = 0 \implies a = 3 \]

From \( (b - 2)^2 = 0 \), we find:

\[ b - 2 = 0 \implies b = 2 \]

### Step 7: Find \( a + b \)
Now, we can find \( a + b \):

\[ a + b = 3 + 2 = 5 \]

Thus, the value of \( a + b \) is:

\[ \boxed{5} \]