Question

value: 3
solve for x. (remember to check for extraneous solutions.)
\\(\sqrt{8x - 12} = x\\)
\\(\circ\\) a. no solution
\\(\circ\\) b. \\(x=2\\)
\\(\circ\\) c. \\(x=6\\)
\\(\circ\\) d. \\(x=6\\) and \\(x=2\\)

Explanation:

Step1: Square both sides to eliminate root

$$(\sqrt{8x-12})^2 = x^2$$
$$8x - 12 = x^2$$

Step2: Rearrange to quadratic form

$$x^2 - 8x + 12 = 0$$

Step3: Factor the quadratic equation

$$(x-2)(x-6) = 0$$

Step4: Solve for x values

$$x-2=0 \implies x=2$$
$$x-6=0 \implies x=6$$

Step5: Check x=2 in original equation

$$\sqrt{8(2)-12} = \sqrt{16-12} = \sqrt{4} = 2$$
This matches x=2, so valid.

Step6: Check x=6 in original equation

$$\sqrt{8(6)-12} = \sqrt{48-12} = \sqrt{36} = 6$$
This matches x=6, so valid.

Answer:

d. x=6 and x=2