Question

kris solved the radical equation \\(\sqrt{15 - x} + x = 3\\) and found that \\(x = -1\\) or \\(x = 6\\). which statement correctly describes the solution set of the radical equation? (1 point) \\(\bigcirc\\ x = -1\\) or \\(x = -6\\) \\(\bigcirc\\ x = -1\\) or \\(x = 6\\) \\(\bigcirc\\ x = -1\\) \\(\bigcirc\\ x = 1\\)

Explanation:

Step1: Isolate the radical term

$\sqrt{15 - x} = 3 - x$

Step2: Square both sides

$(\sqrt{15 - x})^2 = (3 - x)^2$
$15 - x = 9 - 6x + x^2$

Step3: Rearrange to quadratic form

$x^2 - 5x - 6 = 0$

Step4: Factor the quadratic

$(x - 6)(x + 1) = 0$
$x = 6$ or $x = -1$

Step5: Verify solutions in original equation

For $x=6$: $\sqrt{15 - 6} + 6 = 3 + 6 = 9
eq 3$, so $x=6$ is extraneous.
For $x=-1$: $\sqrt{15 - (-1)} + (-1) = 4 - 1 = 3$, which matches the original equation.

Answer:

$\boldsymbol{x=-1}$ (corresponding to the option: $\boldsymbol{x=-1}$)