Questions: Does x^2=-9 have a real number solution? Why or why not?

Solution Steps

To determine if \( x^2 = -9 \) has a real number solution, we need to consider the properties of real numbers. Specifically, the square of any real number is always non-negative. Therefore, \( x^2 = -9 \) cannot have a real number solution because the left-hand side is non-negative while the right-hand side is negative.

We start with the equation \( x^2 = -9 \). To determine if there is a real number solution, we need to consider the properties of real numbers.

For any real number \( x \), \( x^2 \) is always non-negative. This means \( x^2 \geq 0 \) for all \( x \in \mathbb{R} \).

In the given equation \( x^2 = -9 \), the left-hand side \( x^2 \) is non-negative, while the right-hand side is \(-9\), which is negative.

Since a non-negative number cannot equal a negative number, the equation \( x^2 = -9 \) has no real number solutions.

Final Answer