To solve the equation [tex]\(x^2 = -2\)[/tex], we can follow these steps:
1. Identify the equation: The given equation is [tex]\(x^2 = -2\)[/tex].
2. Take the square root of both sides: To solve for [tex]\(x\)[/tex], we need to take the square root of both sides of the equation. Taking the square root on the left-hand side, we get:
[tex]\[
\sqrt{x^2} = \sqrt{-2}
\][/tex]
This simplifies to:
[tex]\[
x = \pm \sqrt{-2}
\][/tex]
3. Simplify the square root of a negative number: The square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i^2 = -1\)[/tex]. So, we can rewrite [tex]\(\sqrt{-2}\)[/tex] as:
[tex]\[
\sqrt{-2} = \sqrt{2} \cdot i
\][/tex]
4. Include the solution with both positive and negative roots: Since taking the square root yields both a positive and a negative solution, we have:
[tex]\[
x = \pm \sqrt{2} \cdot i
\][/tex]
Thus, the solutions to the equation [tex]\(x^2 = -2\)[/tex] are:
[tex]\[
x = \pm \sqrt{2} i
\][/tex]
So, filling in the blank:
[tex]\[
\pm \sqrt{2} i
\][/tex]
