To solve the problem, we start by calculating [tex]\( x^2 \)[/tex] where [tex]\( x = -2i \sqrt{5} \)[/tex].
1. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[
x = -2i \sqrt{5}
\][/tex]
[tex]\[
x^2 = (-2i \sqrt{5})^2
\][/tex]
2. Simplify [tex]\( (-2i \sqrt{5})^2 \)[/tex]:
[tex]\[
(-2i \sqrt{5})^2 = (-2i)^2 \cdot (\sqrt{5})^2
\][/tex]
[tex]\[
(-2i)^2 = (-2)^2 \cdot (i)^2 = 4 \cdot (-1) = -4
\][/tex]
[tex]\[
(\sqrt{5})^2 = 5
\][/tex]
[tex]\[
-4 \cdot 5 = -20
\][/tex]
Therefore,
[tex]\[
x^2 = -20
\][/tex]
3. Formulate the original equation using [tex]\( x^2 \)[/tex]:
We know that [tex]\( x^2 + \square = 0 \)[/tex]. From the calculation above:
[tex]\[
x^2 = -20
\][/tex]
To satisfy this, we can write:
[tex]\[
x^2 + 20 = 0
\][/tex]
Therefore, the missing number is [tex]\( 20 \)[/tex].
4. Find the two solutions of the equation:
The given equation [tex]\( x^2 + 20 = 0 \)[/tex] can be written and solved as:
[tex]\[
x^2 = -20
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm \sqrt{-20} = \pm \sqrt{-1 \cdot 20} = \pm \sqrt{-1} \cdot \sqrt{20} = \pm i \cdot \sqrt{20}
\][/tex]
We simplify [tex]\( \sqrt{20} \)[/tex] further:
[tex]\[
\sqrt{20} = \sqrt{4 \cdot 5} = 2 \sqrt{5}
\][/tex]
Hence,
[tex]\[
x = \pm 2i \sqrt{5}
\][/tex]
So, the two solutions to the equation are:
[tex]\[
x = -2i \sqrt{5} \quad \text{and} \quad x = 2i \sqrt{5}
\][/tex]
Thus, rewriting the final answers:
- The missing number: [tex]\(\boxed{20}\)[/tex]
- The two solutions in simplified, rationalized form are:
[tex]\[
x = -2i \sqrt{5} \quad \text{and} \quad x = 2i \sqrt{5}
\][/tex]
