To solve the quadratic equation [tex]\( x^2 + 121 = 0 \)[/tex], let's follow a step-by-step approach:
1. Rewrite the equation:
[tex]\[
x^2 + 121 = 0
\][/tex]
2. Isolate [tex]\( x^2 \)[/tex] by subtracting [tex]\( 121 \)[/tex] from both sides:
[tex]\[
x^2 = -121
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm\sqrt{-121}
\][/tex]
4. Simplify the expression under the square root:
Notice that [tex]\( -121 \)[/tex] can be written as [tex]\( -1 \times 121 \)[/tex]. Therefore,
[tex]\[
\sqrt{-121} = \sqrt{-1 \times 121} = \sqrt{-1} \times \sqrt{121}
\][/tex]
5. Use the fact that [tex]\( \sqrt{-1} \)[/tex] is defined as [tex]\( i \)[/tex] (the imaginary unit):
[tex]\[
\sqrt{-121} = i \times 11 = 11i
\][/tex]
6. So, the solutions will be:
[tex]\[
x = \pm 11i
\][/tex]
Thus, the solutions to the equation [tex]\( x^2 + 121 = 0 \)[/tex] are [tex]\( 11i \)[/tex] and [tex]\( -11i \)[/tex], which corresponds to option:
D. [tex]\( -11i, 11i \)[/tex]
