Questions: Solve the radical equation. Check all proposed solutions. x - sqrt(4x - 19) = 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 ∅.

Solution Steps

Start with the original equation: \[ x - \sqrt{4x - 19} = 6 \] Isolate the radical term: \[ -\sqrt{4x - 19} = 6 - x \] Multiply both sides by -1: \[ \sqrt{4x - 19} = x - 6 \]

Square both sides to eliminate the radical: \[ 4x - 19 = (x - 6)^2 \] Expand the right side: \[ 4x - 19 = x^2 - 12x + 36 \]

Rearrange the equation to set it to zero: \[ 0 = x^2 - 12x + 36 - 4x + 19 \] Combine like terms: \[ 0 = x^2 - 16x + 55 \]

Factor the quadratic equation: \[ 0 = (x - 11)(x - 5) \] Set each factor to zero: \[ x - 11 = 0 \quad \text{or} \quad x - 5 = 0 \] Thus, the solutions are: \[ x = 11 \quad \text{and} \quad x = 5 \]

Substitute \( x = 11 \) back into the original equation: \[ 11 - \sqrt{4(11) - 19} = 6 \] Calculate: \[ 11 - \sqrt{44 - 19} = 6 \quad \Rightarrow \quad 11 - \sqrt{25} = 6 \quad \Rightarrow \quad 11 - 5 = 6 \] This is valid.

Now check \( x = 5 \): \[ 5 - \sqrt{4(5) - 19} = 6 \] Calculate: \[ 5 - \sqrt{20 - 19} = 6 \quad \Rightarrow \quad 5 - \sqrt{1} = 6 \quad \Rightarrow \quad 5 - 1 = 6 \] This is not valid.

The only valid solution is: \[ \{ 11 \} \]

Final Answer