What is the value of (a - b) when ` a^(2) + b^(2) - 6 a - 6 b + 18 = 0 ` ?

Options

A. 0
B. 3
C. 6
D. 9

Answer

To find the value of \( a - b \) given the equation:

\[ a^2 + b^2 - 6a - 6b + 18 = 0 \]

we can follow these steps:

### Step 1: Rearranging the equation
We start with the equation:

\[ a^2 + b^2 - 6a - 6b + 18 = 0 \]

### Step 2: Completing the square for \( a \)
We can complete the square for the \( a \) terms:

\[ a^2 - 6a = (a - 3)^2 - 9 \]

### Step 3: Completing the square for \( b \)
Now, we complete the square for the \( b \) terms:

\[ b^2 - 6b = (b - 3)^2 - 9 \]

### Step 4: Substitute back into the equation
Substituting these completed squares back into the equation gives us:

\[ ((a - 3)^2 - 9) + ((b - 3)^2 - 9) + 18 = 0 \]

### Step 5: Simplifying the equation
Now we simplify:

\[ (a - 3)^2 + (b - 3)^2 - 9 - 9 + 18 = 0 \]
\[ (a - 3)^2 + (b - 3)^2 = 0 \]

### Step 6: Analyzing the equation
Since the sum of two squares is zero, both squares must individually be zero:

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

### Step 7: Solving for \( a \) and \( b \)
From these equations, we find:

\[ a - 3 = 0 \quad \Rightarrow \quad a = 3 \]
\[ b - 3 = 0 \quad \Rightarrow \quad b = 3 \]

### Step 8: Finding \( a - b \)
Now we can find \( a - b \):

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

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

\[ \boxed{0} \]

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