Question

solve the radical equation. check all proposed solutions.\\(\sqrt{x + 13} = x - 7\\)\\(\\)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.

Explanation:

Step1: Square both sides to eliminate radical

$$(\sqrt{x+13})^2=(x-7)^2$$
$$x+13=x^2-14x+49$$

Step2: Rearrange to quadratic equation

$$x^2-15x+36=0$$

Step3: Factor the quadratic

$$(x-12)(x-3)=0$$

Step4: Solve for x values

$$x-12=0 \implies x=12$$
$$x-3=0 \implies x=3$$

Step5: Check x=3 in original equation

$$\sqrt{3+13}=3-7$$
$$\sqrt{16}=-4$$

$$4 eq -4 \text{ (extraneous solution)}$$

Step6: Check x=12 in original equation

$$\sqrt{12+13}=12-7$$
$$\sqrt{25}=5$$
$$5=5 \text{ (valid solution)}$$

Answer:

A. The solution set is {12}