Cyrus solved a quadratic equation. His work is shown below. In which step did Cyrus make an error?

[tex]\[
\begin{array}{l}
\frac{1}{3}(x-1)^2=12 \\
(x-1)^2=36 \quad \text{Step 1} \\
x-1=6 \quad \text{Step 2} \\
x=7 \quad \text{Step 3} \\
\end{array}
\][/tex]



Answer :

Let's carefully analyze the problem Cyrus worked through:

1. Starting Equation:
[tex]\[ \frac{1}{3}(x-1)^2 = 12 \][/tex]
To eliminate the fraction, multiply both sides of the equation by 3:
[tex]\[ (x-1)^2 = 36 \quad \text{(Step 1)} \][/tex]
This step is correct.

2. Taking the Square Root:
The next step is where Cyrus encounters an error. When solving [tex]\((x-1)^2 = 36\)[/tex], we must consider both the positive and negative square roots:
[tex]\[ x - 1 = 6 \quad \text{or} \quad x - 1 = -6 \][/tex]
However, Cyrus wrote only:
[tex]\[ x - 1 = 6 \quad \text{(Step 2)} \][/tex]
Missing the second potential solution [tex]\((x - 1 = -6)\)[/tex] is the mistake.

3. Solving for [tex]\( x \)[/tex]:
From [tex]\(x - 1 = 6\)[/tex]:
[tex]\[ x = 6 + 1 = 7 \quad \text{(Step 3)} \][/tex]
This solution is correct but incomplete.

4. Finding the Second Solution:
We also need to consider:
[tex]\[ x - 1 = -6 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = -6 + 1 = -5 \][/tex]

Thus, the two solutions to the equation [tex]\(\frac{1}{3}(x-1)^2 = 12\)[/tex] are:
[tex]\[ x = 7 \quad \text{or} \quad x = -5 \][/tex]

Error Identification:
- Cyrus made an error in Step 2 by not considering both potential solutions [tex]\( x - 1 = 6 \)[/tex] and [tex]\( x - 1 = -6 \)[/tex].

To summarize, the error occurred in Step 2 because Cyrus did not account for both the positive and negative square roots when solving the equation. The correct solutions are [tex]\( x = 7 \)[/tex] and [tex]\( x = -5 \)[/tex].