To solve the equation [tex]\( x^2 = -81 \)[/tex], let's analyze it step by step.
1. Understanding the equation: We have [tex]\( x^2 = -81 \)[/tex]. This equation states that the square of some number [tex]\( x \)[/tex] equals -81.
2. Properties of squares: One important property to recall is that the square of any real number is always non-negative. In other words, when you square any real number, the result cannot be negative.
3. Check for real solutions:
- For [tex]\( x^2 \)[/tex] to be -81, [tex]\( x \)[/tex] itself would have to be some value that, when squared, gives -81.
- However, it is impossible for the square of a real number to result in a negative value, because:
- Squaring a positive number yields a positive result.
- Squaring a negative number also yields a positive result.
- Squaring zero yields zero, which is non-negative.
Since no real number squared can be negative, there are no real solutions to the equation [tex]\( x^2 = -81 \)[/tex].
Conclusion: Therefore, the correct answer is [tex]\( F. \text{None} \)[/tex].
