\( 1 \leftarrow \quad \) Solve the following equation by the square root method. \[ (x-6)^{2}=9 \]

To solve the equation \( (x - 6)^{2} = 9 \) using the **square root method**, follow these steps: ### Step 1: Take the Square Root of Both Sides Apply the square root to both sides of the equation. Remember that taking the square root introduces both the positive and negative roots. \[ \sqrt{(x - 6)^{2}} = \sqrt{9} \] ### Step 2: Simplify Both Sides The square root of a squared term simplifies to the absolute value of the original expression. Also, calculate the square root of 9. \[ |x - 6| = 3 \] ### Step 3: Remove the Absolute Value The equation \( |x - 6| = 3 \) implies two possible scenarios: 1. \( x - 6 = 3 \) 2. \( x - 6 = -3 \) ### Step 4: Solve for \( x \) in Each Scenario **Scenario 1: \( x - 6 = 3 \)** \[ x = 3 + 6 \\ x = 9 \] **Scenario 2: \( x - 6 = -3 \)** \[ x = -3 + 6 \\ x = 3 \] ### Step 5: State the Solutions The equation has two solutions: \[ x = 3 \quad \text{and} \quad x = 9 \] ### Verification To ensure that both solutions are correct, substitute them back into the original equation. **For \( x = 3 \):** \[ (3 - 6)^{2} = (-3)^{2} = 9 \quad \checkmark \] **For \( x = 9 \):** \[ (9 - 6)^{2} = (3)^{2} = 9 \quad \checkmark \] Both values satisfy the original equation.