Question

solve.
\\( x - \sqrt{5 - 2x} = -5 \\)

select the correct choice below and fill in any answer boxes present in your choice.

\\( \bigcirc \\) a. \\( x = \square \\) (simplify your answer. use a comma to separate answers as needed.)
\\( \bigcirc \\) b. there is no solution.

Explanation:

Step1: Isolate the radical term

$x + 5 = \sqrt{5 - 2x}$

Step2: Square both sides

$(x + 5)^2 = (\sqrt{5 - 2x})^2$
$x^2 + 10x + 25 = 5 - 2x$

Step3: Rearrange to quadratic form

$x^2 + 10x + 2x + 25 - 5 = 0$
$x^2 + 12x + 20 = 0$

Step4: Factor the quadratic

$(x + 10)(x + 2) = 0$

Step5: Solve for x

$x + 10 = 0 \implies x = -10$
$x + 2 = 0 \implies x = -2$

Step6: Verify solutions

For $x=-10$: $-10 - \sqrt{5 - 2(-10)} = -10 - \sqrt{25} = -10 - 5 = -15
eq -5$ (extraneous)
For $x=-2$: $-2 - \sqrt{5 - 2(-2)} = -2 - \sqrt{9} = -2 - 3 = -5$ (valid)

Answer:

A. $x = -2$