`2sqrt(x-k)=x-6`
If k=3, what is the solution of the equation above?

Options

A. `{4, 12}`
B. `{3}`
C. `{4}`
D. `{12}`

Answer

To solve the equation \( 2\sqrt{x-k} = x - 6 \) given \( k = 3 \), we will follow these steps:

### Step 1: Substitute the value of \( k \)
Substituting \( k = 3 \) into the equation gives us:
\[ 2\sqrt{x - 3} = x - 6 \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ (2\sqrt{x - 3})^2 = (x - 6)^2 \]
This simplifies to:
\[ 4(x - 3) = (x - 6)(x - 6) \]

### Step 3: Expand both sides
Expanding both sides results in:
\[ 4x - 12 = x^2 - 12x + 36 \]

### Step 4: Rearrange the equation
Now, we will rearrange the equation to set it to zero:
\[ 0 = x^2 - 12x + 36 - 4x + 12 \]
Combining like terms gives us:
\[ 0 = x^2 - 16x + 48 \]

### Step 5: Factor the quadratic equation
Next, we factor the quadratic equation:
\[ 0 = (x - 12)(x - 4) \]

### Step 6: Solve for \( x \)
Setting each factor equal to zero gives us the potential solutions:
\[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \]
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

### Step 7: Check for extraneous solutions
We need to check which of these solutions satisfies the original equation:
1. For \( x = 12 \):
\[ 2\sqrt{12 - 3} = 2\sqrt{9} = 6 \]
\[ 12 - 6 = 6 \quad \text{(True)} \]

2. For \( x = 4 \):
\[ 2\sqrt{4 - 3} = 2\sqrt{1} = 2 \]
\[ 4 - 6 = -2 \quad \text{(False)} \]

Thus, the only valid solution is:
\[ \boxed{12} \]