Question

solve the equation.$sqrt{6x} - x + 12 = 0$select the correct choice below and, if necessary, fill in the answer box to complete your choicea. the solution set is (simplify your answer. type an integer or a fraction. use a comma to separate answers as needed.)b. there is no solution.

Explanation:

Step1: Isolate the radical term

$\sqrt{6x} = x - 12$

Step2: Square both sides to eliminate radical

$(\sqrt{6x})^2 = (x - 12)^2$
$6x = x^2 - 24x + 144$

Step3: Rearrange to quadratic equation

$x^2 - 30x + 144 = 0$

Step4: Factor the quadratic

$(x - 24)(x - 6) = 0$

Step5: Solve for x

$x - 24 = 0 \implies x=24$; $x - 6 = 0 \implies x=6$

Step6: Verify solutions in original equation

For $x=24$: $\sqrt{6\times24} - 24 + 12 = \sqrt{144} -12 = 12-12=0$ (valid)
For $x=6$: $\sqrt{6\times6} -6 +12 = \sqrt{36} +6 =6+6=12
eq0$ (invalid)

Answer:

A. The solution set is {24}