Question

10 multiple answer 2 points
julia correctly solved the equation $x - 1 = \sqrt{2x + 1}$ by squaring both sides, combining like terms, and factoring. which of the following are true statements?
she got an extraneous solution of $x = 4$
she got an extraneous solution of $x = 1$
she got an extraneous solution of $x = 0$
she got a solution of $x = 4$
she got a solution of $x = -4$
she got a solution of $x = 0$

Explanation:

Step1: Square both sides

$$(x - 1)^2 = (\sqrt{2x + 1})^2$$
$$x^2 - 2x + 1 = 2x + 1$$

Step2: Rearrange to quadratic form

$$x^2 - 4x = 0$$

Step3: Factor the quadratic

$$x(x - 4) = 0$$
$$x = 0 \text{ or } x = 4$$

Step4: Check solutions in original equation

For $x=0$: $0 - 1 = \sqrt{2(0)+1} \implies -1 = 1$, false (extraneous)
For $x=4$: $4 - 1 = \sqrt{2(4)+1} \implies 3 = 3$, true (valid solution)

Answer:

She got an extraneous solution of $x = 0$, She got a solution of $x = 4$