Questions: Solve the radical equation. Check all proposed solutions. sqrt(2x+12) = x-6 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (Use a comma to separate answers as needed. Simplify your answer.) B. The solution set is the empty set.

Solution Steps

To solve the radical equation \(\sqrt{2x + 12} = x - 6\), we will first isolate the square root and then square both sides to eliminate it. This will result in a quadratic equation. We will solve the quadratic equation and check each solution in the original equation to ensure they are valid.

Starting with the equation: \[ \sqrt{2x + 12} = x - 6 \] we isolate the square root on one side.

Next, we square both sides to eliminate the square root: \[ 2x + 12 = (x - 6)^2 \] Expanding the right side gives: \[ 2x + 12 = x^2 - 12x + 36 \]

Rearranging the equation leads to: \[ 0 = x^2 - 14x + 24 \] This can be rewritten as: \[ x^2 - 14x + 24 = 0 \]

Factoring the quadratic, we find: \[ (x - 12)(x - 2) = 0 \] Thus, the solutions are: \[ x = 12 \quad \text{and} \quad x = 2 \]

We need to check each solution in the original equation:

1. For \(x = 12\): \[ \sqrt{2(12) + 12} = \sqrt{36} = 6 \quad \text{and} \quad 12 - 6 = 6 \quad \text{(valid)} \]
2. For \(x = 2\): \[ \sqrt{2(2) + 12} = \sqrt{16} = 4 \quad \text{and} \quad 2 - 6 = -4 \quad \text{(not valid)} \]

Final Answer