We are tasked with simplifying the expression:

$\frac{2 \pm \sqrt{-12}}{2}$

Step 1: Simplify $\sqrt{-12}$

We know that $\sqrt{-12}$ involves a negative number under the square root, which introduces the imaginary unit $i$, where $i = \sqrt{-1}$. Thus, we can rewrite:

$\sqrt{-12} = \sqrt{12} \cdot i$

Since $\sqrt{12} = 2\sqrt{3}$, we have:

$\sqrt{-12} = 2\sqrt{3} \cdot i = 2i\sqrt{3}$

Step 2: Substitute into the original expression

Now, substitute $\sqrt{-12} = 2i\sqrt{3}$ back into the expression:

$\frac{2 \pm 2i\sqrt{3}}{2}$

Step 3: Simplify the fraction

We can simplify this expression by dividing both terms in the numerator by 2:

$\frac{2}{2} \pm \frac{2i\sqrt{3}}{2} = 1 \pm i\sqrt{3}$

Final Answer:

The simplified form of the expression is:

$1 \pm i\sqrt{3}$

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Do you want more details on any step or have any questions?

Here are 5 related questions:

1. How do you simplify other expressions with square roots of negative numbers?
2. What is the significance of the imaginary unit $i$?
3. How would the simplification change if $-12$ were positive?
4. Can this result be represented graphically in the complex plane?
5. How do you simplify more complex expressions involving imaginary numbers?

Tip: When simplifying expressions with imaginary numbers, always factor out $i$ to handle negative square roots properly.