To address Eloise's error step by step, let's carefully examine her process and point out where things went wrong.
1. Given Equation:
[tex]\[
\sqrt{-2x + 1} = x + 3
\][/tex]
2. Transformed Equation:
Eloise transformed the equation:
[tex]\[
\sqrt{-2x + 1} - 1 = x + 3 - 1
\][/tex]
[tex]\[
\sqrt{-2x + 1} - 1 = x + 2
\][/tex]
3. Squaring Both Sides:
Eloise then squared both sides:
[tex]\[
(\sqrt{-2x + 1} - 1)^2 = (x + 2)^2
\][/tex]
Expanding both sides:
[tex]\[
(\sqrt{-2x + 1} - 1)^2 = -2x + 1 - 2\sqrt{-2x + 1} + 1
\][/tex]
[tex]\[
(x + 2)^2 = x^2 + 4x + 4
\][/tex]
4. Combining Terms:
[tex]\[
-2x + 1 - 2\sqrt{-2x + 1} + 1 = x^2 + 4x + 4
\][/tex]
5. Identification of the Error:
The key error in Eloise's method is revealed in Step 2. Eloise subtracted 1 from [tex]\(\sqrt{-2x + 1}\)[/tex] before squaring both sides.
This step:
[tex]\[
\sqrt{-2x + 1} - 1 = x + 3 - 1
\][/tex]
[tex]\[
\sqrt{-2x + 1} - 1 = x + 2
\][/tex]
should not have been performed. Instead, the squaring should have been directly applied to the original equation:
[tex]\[
(\sqrt{-2x + 1})^2 = (x + 3)^2
\][/tex]
This would have directly led to the correct transformed equation without introducing the extraneous terms.
To summarize:
The error Eloise made was subtracting 1 before squaring both sides of the equation. Therefore, the correct choice is:
She subtracted 1 before squaring both sides.
