Jedida solved the quadratic equation \( x^{2}-96=4 x \) by factoring. Her work is shown below. At which step, did Jedida first make a mistake, if at all?

Question

\begin{tabular}{|l|l|}
\hline \multicolumn{2}{|c|}{\( x^{2}-96=4 x \)} \\
\hline Step 1: \( x^{2}+4 x-96=0 \) & Add \( 4 x \) to both sides. \\
\hline Step 2: \( (x+12)(x-8)=0 \) & Factor. \\
\hline Step 3: \( x+12=0 \) or \( x-8=0 \) & Use the Zero Product Property. \\
\hline & \( x=-12 \) or \( x=8 \) \\
\hline
\end{tabular}
(1 point)
Step 3; she did not apply the Zero Product Property correctly
Step 2; she did not factor the left side of the equation correctly
Step 1; she added \( 4 x \) to both sides instead of subtracting
She did not make any mistakes.

Answer 1

To analyze Jedida's solution, let's go step by step.

1. Starting from the original equation: \[ x^{2} - 96 = 4x \]

2. In Step 1, Jedida rearranged the equation to: \[ x^{2} + 4x - 96 = 0 \] This step is correct because she properly added \( 4x \) to both sides.

3. In Step 2, she factored the left side as: \[ (x+12)(x-8) = 0 \] To check if this factoring is correct, we can expand it: \[ (x + 12)(x - 8) = x^2 - 8x + 12x - 96 = x^2 + 4x - 96 \] Since this is equal to the expression in Step 1 (\( x^2 + 4x - 96 \)), her factoring is also correct.

4. In Step 3, she applies the Zero Product Property correctly, which gives us: \[ x + 12 = 0 \quad \text{or} \quad x - 8 = 0 \]

5. Solving these equations yields: \[ x = -12 \quad \text{or} \quad x = 8 \]

After reviewing each step, it is clear that Jedida made no mistakes in her work. Every step logically follows from the previous one, and the results are accurate.

Thus, the correct answer is: She did not make any mistakes.