To solve the quadratic equation [tex]\( x^2 + 10 = 0 \)[/tex], let's follow the steps:
1. Isolate the quadratic term: We need to isolate [tex]\( x^2 \)[/tex] on one side of the equation. We can do this by subtracting 10 from both sides:
[tex]\[
x^2 + 10 - 10 = 0 - 10
\][/tex]
Simplifying this, we get:
[tex]\[
x^2 = -10
\][/tex]
2. Take the square root of both sides: To solve for [tex]\( x \)[/tex], we take the square root of both sides of the equation. Remember, when taking the square root of a negative number, we use the imaginary unit [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex]:
[tex]\[
x = \pm \sqrt{-10}
\][/tex]
3. Simplify the square root: The square root of [tex]\(-10\)[/tex] can be written using the imaginary unit [tex]\( i \)[/tex]:
[tex]\[
x = \pm \sqrt{10} \cdot \sqrt{-1} = \pm \sqrt{10} i
\][/tex]
Therefore, the solutions to the equation [tex]\( x^2 + 10 = 0 \)[/tex] are:
[tex]\[
x = \pm \sqrt{10} i
\][/tex]
By examining the given answer choices, the correct one is:
C. [tex]\( x = \pm \sqrt{10} i \)[/tex]
