To solve the quadratic equation [tex]\( x^2 + 10 = 0 \)[/tex], follow these steps:
1. Isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[
x^2 + 10 = 0 \implies x^2 = -10
\][/tex]
2. Recognize that solving [tex]\( x^2 = -10 \)[/tex] involves imaginary numbers since [tex]\( -10 \)[/tex] is negative. The square root of a negative number introduces the imaginary unit [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex].
3. Express the square root of -10 in terms of [tex]\( i \)[/tex]:
[tex]\[
x^2 = -10 \implies x = \pm \sqrt{-10}
\][/tex]
By definition, [tex]\( \sqrt{-10} \)[/tex] can be written as [tex]\( \sqrt{10} \cdot \sqrt{-1} = \sqrt{10} \cdot i \)[/tex].
4. Determine the solutions for [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt{10} i
\][/tex]
So the solutions to the quadratic equation [tex]\( x^2 + 10 = 0 \)[/tex] are [tex]\( x = \pm \sqrt{10} i \)[/tex].
Therefore, the correct answer is:
C. [tex]\( x = \pm \sqrt{10} i \)[/tex]
