To solve the equation \( x^2 = 6x - 12 \), we first rearrange it to standard quadratic form:
\[ x^2 - 6x + 12 = 0 \]
Next, we can use the quadratic formula, which is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, \( a = 1 \), \( b = -6 \), and \( c = 12 \).
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-6)^2 - 4(1)(12) = 36 - 48 = -12 \]
Since the discriminant is negative, this means the solutions will be complex. Now we apply the quadratic formula:
\[ x = \frac{-(-6) \pm \sqrt{-12}}{2(1)} = \frac{6 \pm \sqrt{-12}}{2} \]
We can simplify \( \sqrt{-12} \):
\[ \sqrt{-12} = \sqrt{12}i = 2\sqrt{3}i \]
Now substitute this back into the formula:
\[ x = \frac{6 \pm 2\sqrt{3}i}{2} = 3 \pm \sqrt{3}i \]
Now let's analyze the students' solutions:
- Student 1: \( x = -3 \pm \sqrt{21} \) (incorrect)
- Student 2: \( x = 3 \pm i\sqrt{3} \) (this is correct, as \( i\sqrt{3} \) corresponds to the calculated \( \sqrt{-12} \))
- Student 3: \( x = 3 \pm \sqrt{21} \) (incorrect)
Therefore, Student 2 found the correct solutions.
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