Question

solve the radical equation. check all proposed solutions.
\\(sqrt{3x} + 8 = x + 2\\)

select the correct choice below and, if necessary, fill in the answer box to comp
a. the solution set is \\(\\{\\ \\}\\).
(use a comma to separate answers as needed. simplify your answer.)
b. the solution set is \\(\varnothing\\).

Explanation:

Step1: Isolate the radical term

Subtract 8 from both sides of the equation \(\sqrt{3x}+8 = x + 2\) to get \(\sqrt{3x}=x + 2-8\), which simplifies to \(\sqrt{3x}=x - 6\).

Step2: Square both sides

Square both sides of the equation \(\sqrt{3x}=x - 6\) to eliminate the square root. We get \((\sqrt{3x})^2=(x - 6)^2\), which simplifies to \(3x=x^{2}-12x + 36\).

Step3: Rearrange into quadratic equation

Rearrange the equation \(3x=x^{2}-12x + 36\) to standard quadratic form \(ax^{2}+bx + c = 0\). Subtract \(3x\) from both sides: \(x^{2}-15x + 36 = 0\).

Step4: Solve the quadratic equation

Factor the quadratic equation \(x^{2}-15x + 36 = 0\). We need two numbers that multiply to 36 and add to - 15. The numbers are - 3 and - 12. So, \((x - 3)(x - 12)=0\). Setting each factor equal to zero gives \(x - 3=0\) or \(x - 12=0\), so \(x = 3\) or \(x = 12\).

Step5: Check the solutions

• Check \(x = 3\): Substitute \(x = 3\) into the original equation \(\sqrt{3x}+8=x + 2\). Left - hand side: \(\sqrt{3\times3}+8=\sqrt{9}+8 = 3 + 8=11\). Right - hand side: \(3 + 2 = 5\). Since \(11

eq5\), \(x = 3\) is an extraneous solution.

• Check \(x = 12\): Substitute \(x = 12\) into the original equation. Left - hand side: \(\sqrt{3\times12}+8=\sqrt{36}+8 = 6 + 8=14\). Right - hand side: \(12 + 2 = 14\). Since \(14 = 14\), \(x = 12\) is a valid solution.

Answer:

12