To solve the equation \( x^2 + 6 = 0 \), follow these steps:
1. Isolate the quadratic term:
[tex]\[
x^2 + 6 = 0
\][/tex]
Subtract 6 from both sides to isolate the \( x^2 \) term:
[tex]\[
x^2 = -6
\][/tex]
2. Take the square root of both sides:
To solve for \( x \), take the square root of both sides of the equation. Remember to include both the positive and negative roots:
[tex]\[
x = \pm \sqrt{-6}
\][/tex]
3. Simplify the expression involving the square root of a negative number:
Recall that the square root of a negative number involves the imaginary unit \( i \), where \( i = \sqrt{-1} \):
[tex]\[
\sqrt{-6} = \sqrt{-1 \cdot 6} = \sqrt{-1} \cdot \sqrt{6} = i \sqrt{6}
\][/tex]
4. Write the final solutions:
Considering both the positive and negative roots, we have:
[tex]\[
x = \pm i \sqrt{6}
\][/tex]
Hence, the solutions to the equation \( x^2 + 6 = 0 \) are:
[tex]\[
x = -i \sqrt{6}, \quad x = i \sqrt{6}
\][/tex]
