The equation \( x^2 - 8 + 12 = 0 \) seems to have a constant term, but no linear term. Let's first simplify it to standard form:

\[
x^2 - 8 + 12 = 0
\]

\[
x^2 + 4 = 0
\]

Now, to solve for \( x \), we rearrange the equation:

\[
x^2 = -4
\]

Next, take the square root of both sides. Since the square root of a negative number involves an imaginary number, we have:

\[
x = \pm \sqrt{-4}
\]

\[
x = \pm 2i
\]

where \( i \) is the imaginary unit, representing \(\sqrt{-1}\).

Therefore, the solutions to the equation are \( x = 2i \) and \( x = -2i \).

Would you like further details on these steps or have any other questions?

### Related Questions

1. What is the imaginary unit \( i \), and how is it used in mathematics?
2. How do you simplify expressions involving imaginary numbers?
3. Can complex numbers be plotted on a graph, and if so, how?
4. What is the difference between real and imaginary numbers?
5. How can we solve quadratic equations with complex solutions?
6. Are there any real-world applications of complex numbers?
7. What is the complex conjugate, and how is it useful in calculations?
8. How does the presence of complex roots affect the graph of a quadratic function?

**Tip:** When solving equations involving imaginary numbers, always remember that the square root of a negative number introduces the imaginary unit \( i \).

Learn how to solve the quadratic equation x^2 - 8 + 12 = 0 with complex solutions involving imaginary numbers. This problem explores key algebraic concepts, including the quadratic equation solving and the usage of the imaginary unit. Suitable for students in Grades 10-12.

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