Question

Students solved the equation x2=6x−12
and their solutions are shown. Which student found the correct solutions?

Student 1: x=−3±21−−√


Student 2: x=3±i3–√


Student 3: x=3±21−−√


(1 point)

Answer 1

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.