To solve the equation [tex]\( 2x^2 + 36 = 0 \)[/tex], we will proceed with a detailed, step-by-step approach:
1. Start with the given equation:
[tex]\[
2x^2 + 36 = 0
\][/tex]
2. Isolate the [tex]\( x^2 \)[/tex] term by subtracting 36 from both sides:
[tex]\[
2x^2 = -36
\][/tex]
3. Divide both sides of the equation by 2 to solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = -18
\][/tex]
4. Recognize that the right side is negative, indicating complex solutions. Now, take the square root of both sides:
[tex]\[
x = \pm \sqrt{-18}
\][/tex]
5. Rewrite the square root of a negative number using the imaginary unit [tex]\( i \)[/tex] ([tex]\( \sqrt{-1} = i \)[/tex]):
[tex]\[
x = \pm \sqrt{18} \cdot i
\][/tex]
6. Simplify [tex]\( \sqrt{18} \)[/tex]. Note that 18 can be factored as [tex]\( 9 \times 2 \)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
\][/tex]
7. Combine this result with the factor [tex]\( i \)[/tex]:
[tex]\[
x = \pm 3\sqrt{2} \cdot i
\][/tex]
So, the solutions to the equation [tex]\( 2x^2 + 36 = 0 \)[/tex] are [tex]\( x = \pm 3i\sqrt{2} \)[/tex].
Therefore, the correct option is:
D. [tex]\( x = \pm 3i\sqrt{2} \)[/tex]
