Question

solve the radical equation. check all proposed solutions.
\sqrt{5x + 4} = x - 4

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

$$(\sqrt{5x+4})^2=(x-4)^2$$
$$5x+4=x^2-8x+16$$

Step2: Rearrange into quadratic equation

$$x^2-13x+12=0$$

Step3: Factor the quadratic

$$(x-1)(x-12)=0$$

Step4: Solve for x

$$x=1 \text{ or } x=12$$

Step5: Check x=1 in original equation

Left side: $\sqrt{5(1)+4}=\sqrt{9}=3$
Right side: $1-4=-3$
$3
eq -3$, so $x=1$ is extraneous.

Step6: Check x=12 in original equation

Left side: $\sqrt{5(12)+4}=\sqrt{64}=8$
Right side: $12-4=8$
$8=8$, so $x=12$ is valid.

Answer:

A. The solution set is {12}.