Question

Hyung-eun is solving the quadratic equation x2−4x−12=0 by grouping. Her work is shown below. At which step does Hyung-eun first make an error?

Answer 1

To determine where Hyung-eun may have made an error in solving the quadratic equation \( x^2 - 4x - 12 = 0 \) by grouping, we can follow the steps typically taken in this method.

1. **Rearrange the equation:** The equation is already set to \( x^2 - 4x - 12 = 0 \).

2. **Factor the quadratic expression:** To use grouping, the expression needs to be rewritten. The first step is to find two numbers that multiply to \(-12\) (the constant term) and add to \(-4\) (the coefficient of \(x\)). The factors of \(-12\) that work are \(2\) and \(-6\) since \(2 + (-6) = -4\).

3. **Rewrite the expression using these factors:**
\[
x^2 - 6x + 2x - 12 = 0
\]
Grouping the terms:
\[
(x^2 - 6x) + (2x - 12) = 0
\]

4. **Factor by grouping:** Now factor out common terms from each group.
\[
x(x - 6) + 2(x - 6) = 0
\]

5. **Combine the factored groups:**
\[
(x - 6)(x + 2) = 0
\]

6. **Solve for \( x \):** Set each factor to zero:
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]

If Hyung-eun follows through these steps, there seems to be no error up to this point. However, if you can provide her actual steps as she wrote them out, then I can pinpoint the exact moment in her process where she made an error.