Question

solve this equation for x: (sqrt{3x + 7} - x = 3). select the correct answer. (\bigcirc) (x = 1, 2) (\bigcirc) (x = -2, -1) (\bigcirc) (x = 2, 8) (\bigcirc) (x = -8, -2)

Explanation:

Step1: Isolate the square root

$\sqrt{3x + 7} = x + 3$

Step2: Square both sides

$(\sqrt{3x + 7})^2 = (x + 3)^2$
$3x + 7 = x^2 + 6x + 9$

Step3: Rearrange to quadratic form

$x^2 + 6x + 9 - 3x - 7 = 0$
$x^2 + 3x + 2 = 0$

Step4: Factor the quadratic

$(x + 1)(x + 2) = 0$

Step5: Find potential solutions

$x + 1 = 0 \implies x = -1$
$x + 2 = 0 \implies x = -2$

Step6: Verify solutions

For $x=-1$: $\sqrt{3(-1)+7} - (-1) = \sqrt{4} + 1 = 2 + 1 = 3$ (valid)
For $x=-2$: $\sqrt{3(-2)+7} - (-2) = \sqrt{1} + 2 = 1 + 2 = 3$ (valid)

Answer:

B. $x = -2, -1$